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Success as an Insurance Agent

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					Random Variables
Discrete Random Variables
Continuous Random Variables
Discrete Probability Distributions
Expected Value for a Discrete Random Variables
Variance and Standard Deviation for a Discrete Random Variables
Binomial Experiment
Binomial Probabilities
Expected Value and Variance and Standard Deviation for a Binomial Distribution
POISSON Distibution
Hypergeometric Distribution
                  Experiment of tossing coin three times
Steps in Experiment                            3
Outcomes Step 1                                2
Outcomes Step 2                                2
Outcomes Step 3                                2
Total Experimental Outcomes                    8 (Sample Points)

P(for each) =                                 0.125 Classical


# SP                                 Toss 1           Toss 2       Toss 3                       Sample Space
                                 1   H                H            H                            (H, H, H)
                                 2   T                H            H                            (T, H, H)
                                 3   H                T            H                            (H, T, H)
                                 4   H                H            T                            (H, H, T)
                                 5   T                T            H                            (T, T, H)
                                 6   T                H            T                            (T, H, T)
                                 7   H                T            T                            (H, T, T)
                                 8   T                T            T                            (T, T, T)

x = Discrete Random Variable =       # of heads in 3 flips
                                                                                                Steps:
X                                    Frequency        P(X) = f(x) Each f(x)>=0?
                                 0                1        0.125       TRUE
                                 1                3        0.375       TRUE
                                 2                3        0.375       TRUE
                                 3                1        0.125       TRUE
                                                  8              1

P(X =1)                                                                                         Experiment of tossing coin three
P(X > = 2)                                                                                                  times
P(x = 0 or x = 3)                                           0.25                                           0.375
P(X not 2)                                                 0.625
                                                                                  Probability




                                                                                                  0.125
.


                                                                                                    0          1
                                                                                                          # of heads in 3 flips
              Success =
              Heads
              Notation for Heads
              is H
              H
              Probability of       # of heads for sample
              Sample Point         points
                             0.125                      3
                             0.125                      2
                             0.125                      2
                             0.125                      2
                             0.125                      1
                             0.125                      1
                             0.125                      1
                             0.125                      0
                                 1

              #1                     Define Random Variable
              #2                     Build Frequency Distribution
              #3                     Calculate Relative Frequency - P(x) = f(x)
              #4                     Check Requirement #1: f(x) >= 0
              #5                     Check Requirement #2: Σf(x) = 1
              #6                     Create a Column Chart for a Discrete Variable (Columns do not touch)
              #7                     Make predictions

riment of tossing coin three
        times
      0.375        0.375



                             0.125



                    2         3
     # of heads in 3 flips
                  Experiment of tossing coin three times
Steps in Experiment                            3
Outcomes Step 1                                2
Outcomes Step 2                                2
Outcomes Step 3                                2
Total Experimental Outcomes                    8 (Sample Points)

P(for each) =                                 0.125 Classical


# SP                                 Toss 1            Toss 2                Toss 3        Sample Space
                                 1   H                 H                     H             (H, H, H)
                                 2   T                 H                     H             (T, H, H)
                                 3   H                 T                     H             (H, T, H)
                                 4   H                 H                     T             (H, H, T)
                                 5   T                 T                     H             (T, T, H)
                                 6   T                 H                     T             (T, H, T)
                                 7   H                 T                     T             (H, T, T)
                                 8   T                 T                     T             (T, T, T)

x = Discrete Random Variable =       # of heads in 3 flips                Experiment of tossing coin three
                                                                                      timesSteps:
X                                    Frequency                              0.375
                                                       P(X) = f(x) Each f(x)>=0?       0.375
                                 0                1         0.125       TRUE
                                                            Probability




                                                                  0.125                               0.125
                                 1                3         0.375       TRUE
                                 2                3         0.375       TRUE
                                 3                1         0.125 0 TRUE 1               2                3
                                                  8               1       # of heads in 3 flips

P(X =1)                                       0.125
P(X > = 2)                                       0.5
P(x = 0 or x = 3)                              0.25          0.25
P(X not 2)                                    0.625         0.625
            Success =
            Heads
            Notation for Heads
            is H
            H
            Probability of       # of heads for sample
            Sample Point         points
                           0.125                      3
                           0.125                      2
                           0.125                      2
                           0.125                      2
                           0.125                      1
                           0.125                      1
                           0.125                      1
                           0.125                      0
                               1
oin three
           #1                   Define Random Variable
           #2                   Build Frequency Distribution
           #3                   Calculate Relative Frequency - P(x) = f(x)
     0.125
           #4                   Check Requirement #1: f(x) >= 0
           #5                   Check Requirement #2: Σf(x) = 1
           #6                   Create a Column Chart for a Discrete Variable (Columns do not touch)
           #7                   Make predictions
  Over last 20 days the number of operating rooms used at TG hospital were:        Steps to Build Discrete Probability Distribut
                           On 3 days only 1 were used                              #1       Define Random Variable
                              On 5 days 2 were used                                #2       Build Frequency Distribution
                              On 8 days 3 were used                                #3       Calculate Relative Frequency - P(x
                           On 4 days all (4) were used                             #4       Check Requirement #1: f(x) >= 0
  Create a Probability Distribution that can be used to make predictions in the
                                      future.                                      #5       Check Requirement #2: Σf(x) = 1
                                                                                   #6       Create a Column Chart for a Discr
Discrete Random
Variable = x =                                                                     #7       Make predictions


                            x=#            Relative
                            operating      Frequency P(x)
                            rooms in use   = f(x) =
Frequency (# of days)       for 1 day      Probability    Is each f(x) >= 0?       P(x)     f(x)
                                                                                   P()      f()
                                                                                   P()      f()
                                                                                   P()      f()
                                                                                   P()      f()



Reason we do the whole distribution? Because then it is easy to make predictions
                                                                                                    check
                        2   <=2            P(x<=2)         f(1) + f(2)
                        2   <2             P(x<2)          f(1)
                        2   >=2            P(x>=2)         f(2) + f(3) + f(4)
                        2   >2             P(x>2)          f(3) + f(4)
ete Probability Distribution:
ndom Variable
uency Distribution
Relative Frequency - P(x) = f(x)
 uirement #1: f(x) >= 0

uirement #2: Σf(x) = 1
olumn Chart for a Discrete Variable (Columns do not touch)




                         Discrete Probability Charts don't have columns that touch.


             SUMIF function:
             range = range with all criteria
             criteria = criteria such as 2, <=2, >=2, >2
             sum_range = range with values to add
             For more about this function, see this video:
             Excel Magic Trick #203: SUMIF function formula 21 Examples
  Over last 20 days the number of operating rooms used at TG hospital were:        Steps to Build Discrete Probability Distribut
                           On 3 days only 1 were used                              #1       Define Random Variable
                              On 5 days 2 were used                                #2       Build Frequency Distribution
                              On 8 days 3 were used                                #3       Calculate Relative Frequency - P(x
                           On 4 days all (4) were used                             #4       Check Requirement #1: f(x) >= 0
  Create a Probability Distribution that can be used to make predictions in the
                                      future.                                      #5            Check Requirement #2: Σf(x) = 1
                                                                                   #6            Create a Column Chart for a Discr
Discrete Random
Variable = x =               X = # of Op. Rooms used in 1 day                      #7            Make predictions


                             x=#             Relative
                             operating       Frequency P(x)
                             rooms in use    = f(x) =
Frequency (# of days)        for 1 day       Probability      Is each f(x) >= 0?   P(x)          f(x)
                         3               1               0.15           TRUE       P(1)          f(1)
                         5               2               0.25           TRUE       P(2)          f(2)
                         8               3                0.4           TRUE       P(3)          f(3)
                         4               4                0.2           TRUE       P(4)          f(4)
                        20                                  1

Reason we do the whole distribution? Because then it is easy to make predictions
                                                                                                        check
                         2   <=2             P(x<=2)          f(1) + f(2)                  0.4                   0.4
                         2   <2              P(x<2)           f(1)                        0.15                  0.15
                         2   >=2             P(x>=2)          f(2) + f(3) + f(4)          0.85                  0.85
                         2   >2              P(x>2)           f(3) + f(4)                  0.6                   0.6
ete Probability Distribution:                              Over last 20 days the number of operating rooms used at TG
ndom Variable                                                                     hospital were:
uency Distribution
Relative Frequency - P(x) = f(x)                                                                    0.4
 uirement #1: f(x) >= 0



                                             Probability
                                                                               0.25
uirement #2: Σf(x) = 1                                                                                           0.2
                                                  0.15
olumn Chart for a Discrete Variable (Columns do not touch)



                                                               1                 2                   3            4
                                                                            X = # of Op. Rooms used in 1 day




                         Discrete Probability Charts don't have columns that touch.


             SUMIF function:
             range = range with all criteria
             criteria = criteria such as 2, <=2, >=2, >2
             sum_range = range with values to add
             For more about this function, see this video:
             Excel Magic Trick #203: SUMIF function formula 21 Examples
sed at TG
Job
satisfaction       IS Senior          Is each f(x) >= IS Middle          Is each f(x) >=
score              Executives         0?              Executives         0?
               1                 5%                                 4%
               2                 9%                                10%
               3                 3%                                12%
               4                42%                                46%
               5                41%                                28%
Job
satisfaction       IS Senior       Is each f(x) >= IS Middle      Is each f(x) >=
score              Executives      0?              Executives     0?
               1              5%        TRUE                   4%      TRUE
               2              9%        TRUE                  10%      TRUE
               3              3%        TRUE                  12%      TRUE
               4             42%        TRUE                  46%      TRUE
               5             41%        TRUE                  28%      TRUE
                            100%                             100%

                       Discrete Probability Distribution
                      IS Senior Executives       IS Middle Executives
                                                       42%46%           41%
                                                                              28%

                         9% 10%              12%
         5% 4%                          3%

           1                2                3             4              5
                                Job satisfaction score
Mean and Standard Deviation from Raw Data                       Mean and Standard Deviation for Discrete Pr

                                               x = # operating
                                               rooms in use for
                                               1 day             Frequency (# of days)
Mean                                                           1                            3
STDEV                                                          2                            5
STDEVP                                                         3                            8
Raw Data = Xi       X - mu   (X-mu)^2                          4                            4
                1                                                                          20

                1
                1                              E(x)               Σx*f(x)
                                               SD of Discrete
                                               Random
                2                              Variable           sqrt(Σ(x-mean)^2*f(x))
                2
                2
                2
                2
                3
                3
                3
                3
                3
                3
                3
                3
                4
                4
                4
                4
                             Must Equal Zero
                    sd
                    Count
ndard Deviation for Discrete Probability Distribution
          Relative
          Frequency P(x)
          = f(x) =        E(x)
          Probability     =X*f(x) X - mu                  (X-mu)^2 (X-mu)^2*f(x)
                     0.15
                     0.25
                      0.4
                      0.2
                        1         because you are not     Var

                                                          SD
                                    Not equal to zero
                                     using all the data
                                                 points
Mean and Standard Deviation from Raw Data                            Mean and Standard Deviation for Discrete Pr

                                                    x = # operating
                                                    rooms in use for
                                                    1 day             Frequency (# of days)
Mean                     2.650                                      1                            3
STDEV              0.98808693                                       2                            5
STDEVP             0.96306801                                       3                            8
Raw Data = Xi     X - mu       (X-mu)^2                             4                            4
                1       -1.650          2.7225                                                  20

                1        -1.650           2.7225
                1        -1.650           2.7225    E(x)               Σx*f(x)
                                                    SD of Discrete
                                                    Random
                2         -0.650           0.4225   Variable           sqrt(Σ(x-mean)^2*f(x))
                2         -0.650           0.4225
                2         -0.650           0.4225
                2         -0.650           0.4225
                2         -0.650           0.4225
                3          0.350           0.1225
                3          0.350           0.1225
                3          0.350           0.1225
                3          0.350           0.1225
                3          0.350           0.1225
                3          0.350           0.1225
                3          0.350           0.1225
                3          0.350           0.1225
                4          1.350           1.8225
                4          1.350           1.8225
                4          1.350           1.8225
                4          1.350           1.8225
                            0.00 Must Equal Zero
                    sd                  0.963068
                    Count                      20
ndard Deviation for Discrete Probability Distribution
          Relative
          Frequency P(x)
          = f(x) =        E(x)
          Probability     =X*f(x) X - mu                   (X-mu)^2 (X-mu)^2*f(x)
                     0.15    0.150            -1.650           2.7225     0.408375
                     0.25    0.500            -0.650           0.4225     0.105625
                      0.4    1.200             0.350           0.1225         0.049
                      0.2    0.800             1.350           1.8225        0.3645
                        1    2.650            -0.600
                                   because you are not     Var               0.9275

                                                           SD          0.963068014
                                     Not equal to zero
                                      using all the data




                  2.650
                                                  points




            0.963068014
            0.963068014
            0.981360288   0.9814
Unit Demand = x          Prob = f(x)                (X - Mean)^2               Mean
                   500                  0.1                        84100              790
                   600                 0.15                        36100
                   700                 0.15                         8100
                   800                  0.2                          100
                   900                  0.2                        12100
                  1000                 0.15                        44100
                  1100                 0.05                        96100


Mean                                          = monthly order quantity
SD                                                                       170


Price per unit                 $125.00
Cost per unit                   $60.00
Assumed Units sold                1000
Sales
Expenses
Gross Profit
Unit Demand = x          Prob = f(x)             (X - Mean)^2
                   500                  0.1                      84100
                   600                 0.15                      36100
                   700                 0.15                       8100
                   800                  0.2                        100
                   900                  0.2                      12100
                  1000                 0.15                      44100
                  1100                 0.05                      96100
                                          1
Mean                                   790 = monthly order quantity
SD                                     170                          170


Price per unit                $125.00
Cost per unit                   $60.00
Assumed Units sold                1000
Sales                      $125,000.00
Expenses                    $47,400.00
Gross Profit                $77,600.00
                      Prob. Of State Stock A        Stock B     Stock C
State of Economy      of Economy       Return       Return      Return
Boom                              0.15         0.15        0.25      0.11
Normal                            0.30         0.07        0.13      0.10
Bust                              0.55        -0.02     -0.135       0.03
                      E(Ri)
                      Standard
Sd = Proxy for Risk   Deviation
                      CV = SD/Mean
                      Prob. Of State Stock A        Stock B     Stock C
State of Economy      of Economy       Return       Return      Return
Boom                              0.15         0.15        0.25      0.11
Normal                            0.30         0.07        0.13      0.10
Bust                              0.55        -0.02      -0.135      0.03
                      E(Ri)                 0.0325     0.00225      0.063
                      Standard
Sd = Proxy for Risk   Deviation         0.0633147 0.156409 0.03662
                      CV = SD/Mean      1.9481443 69.51499 0.58126
 pi =                                     0.2
 1 - pi =                                 0.8
 n=                                         4
 Fixed # of Trials?                 Yes
 Each Trial Independent?            Yes
 Success is?                        Sale     S
 Fail is?                           No Sale NS
 Success or fail each trial?        Yes
 pi the same each Trial?            Yes
 Sample Space =                           16

                                                     Build Discrete Probability Distribution with n = 4, pi = 0.2
                                                  Attempted Sale                    # of          Probability        Probability of Occurrence   Probability of    # of Sales (Random
       Possible Outcomes                                                                                                                                                                                            P(x)          P(x)
                                    1st         2nd        3rd       4th           Sales 1st 2nd 3rd 4th                   Calculation           Occurrence             Variable X)
                                1   S           S          S          S                  4   0.2   0.2   0.2   0.2   0.2*0.2*0.2*0.2 =                    0.0016                                              0    P(0) =     0.4096
                                2   S           S          S          NS                 3   0.2   0.2   0.2   0.8   0.2*0.2*0.2*0.8 =                    0.0064                                              1    P(1) =     0.4096
                                3   S           S          NS         S                  3   0.2   0.2   0.8   0.2   0.2*0.2*0.8*0.2 =                    0.0064                                              2    P(2) =     0.1536
                                4   S           NS         S          S                  3   0.2   0.8   0.2   0.2   0.2*0.8*0.2*0.2 =                    0.0064                                              3    P(3) =     0.0256
                                5   NS          S          S          S                  3   0.8   0.2   0.2   0.2   0.8*0.2*0.2*0.2 =                    0.0064                                              4    P(4) =     0.0016
                                6   S           S          NS         NS                 2   0.2   0.2   0.8   0.8   0.2*0.2*0.8*0.8 =                    0.0256                                                              1.0000
                                7   S           NS         NS         S                  2   0.2   0.8   0.8   0.2   0.2*0.8*0.8*0.2 =                    0.0256
                                8   NS          NS         S          S                  2   0.8   0.8   0.2   0.2   0.8*0.8*0.2*0.2 =                    0.0256                                                   E(x)              0.8         0.8
                                9   S           NS         S          NS                 2   0.2   0.8   0.2   0.8   0.2*0.8*0.2*0.8 =                    0.0256                                                   SD                0.8         0.8
                               10   NS          S          NS         S                  2   0.8   0.2   0.8   0.2   0.8*0.2*0.8*0.2 =                    0.0256
                               11   NS          S          S          NS                 2   0.8   0.2   0.2   0.8   0.8*0.2*0.2*0.8 =                    0.0256                                            Build Discrete Probability Distribution with n = 4, pi = 0.2
                               12   S           NS         NS         NS                 1   0.2   0.8   0.8   0.8   0.2*0.8*0.8*0.8 =                    0.1024
                               13   NS          S          NS         NS                 1   0.8   0.2   0.8   0.8   0.8*0.2*0.8*0.8 =                    0.1024




                                                                                                                                                                       P(x) = Probability of X Sales in 4
                               14   NS          NS         S          NS                 1   0.8   0.8   0.2   0.8   0.8*0.8*0.2*0.8 =                    0.1024                                            0.4096          0.4096
                               15   NS          NS         NS         S                  1   0.8   0.8   0.8   0.2   0.8*0.8*0.8*0.2 =                    0.1024
                               16   NS          NS         NS         NS                 0   0.8   0.8   0.8   0.8   0.8*0.8*0.8*0.8 =                    0.4096




                                                                                                                                                                                  Attempts
                                                                                                                                                                                                                                           0.1536

                                                                                                                                                                                                                                                       0.0256        0.0016

                                                                                                                                                                                                               0              1              2           3                 4

                                                                                                                                                                                                                             X Random Variable (# of Sales)




b9673ec4-5698-4e75-9513-72f6628e80d5.xls                                                                                BDPD                                                                                                                                         Page 21 of 57
Binomial Experiment 4 Requirements:
  Experiment consists of a sequence
  of n identical trials. Random
  Variable counts the number of
  successes in a Fixed number of
1 trials, n.
  Fixed # of Identical Trials = n
  Only 2 outcomes are possible on
  each indetical trial. Success or
2 Failure.
  Each trial only results in S or F
  Probability of Success = p = (π "pi").
  Probability of Failure = 1-p.
  Probability remains the same on
3 each trial.
  p remains the same for each trial
  The trials are independent (one
4 does not affact the next)
  All events are independent

   An insurance agent has appointments with 4 clients tomorrow. From past
   data, the chance of making a sale is 1 in 5. What is likelihood that she will                         = p (π) =
                           sell 3 policies in 4 tries?                                       Fixed # of Identical Trials = n =
                                   Variables                                       P(x) = f(x)
                                        Success                                    P(x = 0)
                                        Failure                                    P(x >= 0)
                                        n                                          P(x > 0)
                                        x                                          P(x < 0)
                                        p (π)                                      P(x <= 0)

                                           X        P(x) = f(x)
                                                0
                                                1
                                                2
                                                3
                                                4
             = p (π) =
Fixed # of Identical Trials = n =
    P(x) = f(x)                SUM




                                        #S.P. = COMBIN
                                                                        An insurance agent has appointments with 4
                                        x=
                                                                       clients tomorrow. From past data, the chance
                                        n=
                                                                        of making a sale is 1 in 5. What is likelihood
                                      s p=
                                                                            that she will sell 3 policies in 4 tries?
                                     ns (1-p) =
                                        # S.P.               C1   C2   C3   C4    P1 P2 P3
                                                         1   ns   s    s    s       0.8 0.2 0.2
                                                         2   s    ns   s    s       0.2 0.8 0.2
                                                         3   s    s    ns   s       0.2 0.2 0.8
                                                         4   s    s    s    ns      0.2 0.2 0.2
ent has appointments with 4
 . From past data, the chance
  is 1 in 5. What is likelihood
 sell 3 policies in 4 tries?

             P4 P(S.P.)           Multiply
               0.2
               0.2
               0.2
               0.8
                                  Add
Binomial Experiment 4 Requirements:
  Experiment consists of a sequence
  of n identical trials. Random
  Variable counts the number of
  successes in a Fixed number of
1 trials, n.
  Fixed # of Identical Trials = n          Yes
  Only 2 outcomes are possible on
  each indetical trial. Success or
2 Failure.
  Each trial only results in S or F        Yes
  Probability of Success = p = (π "pi").
  Probability of Failure = 1-p.
  Probability remains the same on
3 each trial.
  p remains the same for each trial        Yes
  The trials are independent (one
4 does not affact the next)
  All events are independent               Yes

  An insurance agent has appointments with 4 clients tomorrow. From past
   data, the chance of making a sale is 1 in 5. What is likelihood that she will             Probability of Sale = p (π) = 0.2
                             sell 3 policies in 4 tries?                                    Fixed # of Identical Trials = n = 4
                                     Variables                                     P(x) = f(x)
  Sale                                    Success s                                P(x = 3)
  Not Sale                                Failure ns                               P(x >= 3)
  Attempts at Sale                        n                                    4   P(x > 3)
  X = # of Sale is made in 4 tries        x                                    3   P(x < 3)
  Probability of Sale                     p (π)                             0.2    P(x <= 3)

                                           X         P(x) = f(x)
                                                 0                      0.4096
                                                 1                      0.4096
                                                 2                      0.1536
                                                 3                      0.0256
                                                 4                      0.0016
              Binomial Distribution, n = 4,
                         p = .2
           0.4096      0.4096
                                    0.1536
                                                 0.0256       0.0016

              0           1            2            3           4
                        X = # of Sale is made in 4 tries

 Probability of Sale = p (π) = 0.2
Fixed # of Identical Trials = n = 4
    P(x) = f(x)                 SUM
                       0.0256       0.0256
                       0.0272       0.0272
                       0.0016       0.0016
                       0.9728       0.9728
                       0.9984       0.9984




                                                           #S.P. = COMBIN           4
                                                                                              An insurance agent has appointments with 4
                                                           x=                       3
                                                                                             clients tomorrow. From past data, the chance
                                                           n=                       4
                                                                                              of making a sale is 1 in 5. What is likelihood
                                                         s p=                     0.2
                                                                                                  that she will sell 3 policies in 4 tries?
                                                        ns (1-p) =                0.8
                                                           # S.P.               C1      C2   C3   C4    P1 P2 P3 P4
                                                                            1   ns      s    s    s       0.8 0.2 0.2 0.2
                                                                            2   s       ns   s    s       0.2 0.8 0.2 0.2
                                                                            3   s       s    ns   s       0.2 0.2 0.8 0.2
                                                                            4   s       s    s    ns      0.2 0.2 0.2 0.8
 s appointments with 4
m past data, the chance
n 5. What is likelihood
 policies in 4 tries?

             P(S.P.)     Multiply
                 0.0064
                 0.0064
                 0.0064
                 0.0064
                 0.0256 Add
            Binomial Experiment 4 Requirements:

       Experiment consists of a sequence of n identical trials.
       Random Variable counts the number of successes in a
   1   Fixed number of trials, n.
       Fixed # of Identical Trials = n
       Only 2 outcomes are possible on each indetical trial.
   2   Success or Failure.
       Each trial only results in S or F
       Probability of Success = p = (π "pi"). Probability of
       Failure = 1-p. Probability remains the same on each
   3   trial.
       p remains the same for each trial
       The trials are independent (one does not affact the
   4   next)
       All events are independent


        A flight from Oakland to Seattle occurs 6 times per day. The probability that
       any 1 flight is late is 10%. What is the probability that exactly 2 planes are late?
                Less than 2 are late? What is the mean and standard deviation?
                                             Variables
                                                                   Success
                                                                   Failure
                                                                   n
                                                                   x
                                                                   p (π)

                                      = p (π) =
                          Fixed # of Identical Trials = n =
       P(x) = f(x)                                                P(x) = f(x)
   2   P(x = 2)
   2   P(x >= 2)
   2   P(x > 2)
   2   P(x < 2)
   2   P(x <= 2)

Mean   Σx*f(x)
SD     sqrt(Σ(x-mean)^2*f(x))
Mean   n*p
SD     sqrt(n*p*(1-p))

                                                                  X             f(x)
                                                                            0      0.531441
                                                                            1      0.354294
                                                                            2      0.098415
                                                                            3        0.01458
4   0.001215
5     5.4E-05
6   0.000001
     A flight from Oakland to Seattle occurs 6 times per day. The probability that any 1 flight is
      late is 10%. What is the probability that exactly 2 planes are late? Less than 2 are late?
                             What is the mean and standard deviation?

0.531


               0.354




                               0.098
                                               0.015            0.001           0.000           0.000

 0                1               2               3               4                5                 6
0   1   2   3   4   5   6
            Binomial Experiment 4 Requirements:

       Experiment consists of a sequence of n identical trials.
       Random Variable counts the number of successes in a
   1   Fixed number of trials, n.
       Fixed # of Identical Trials = n                            Yes
       Only 2 outcomes are possible on each indetical trial.
   2   Success or Failure.
       Each trial only results in S or F                          Yes
       Probability of Success = p = (π "pi"). Probability of
       Failure = 1-p. Probability remains the same on each
   3   trial.
       p remains the same for each trial                          Yes
       The trials are independent (one does not affact the
   4   next)
       All events are independent                                 Yes


         A flight from Oakland to Seattle occurs 6 times per day. The probability that
       any 1 flight is late is 10%. What is the probability that exactly 2 planes are late?
                Less than 2 are late? What is the mean and standard deviation?
                                             Variables
       Late Plane                                                  Success     l
       Not Late Plane                                              Failure     nl
       # of flights per day                                        n                      6
       # successes = # of Late Planes                              x                      2
       Probability of Late Plane                                   p (π)                0.1

                      Probability of Late Plane = p (π) = 0.1
                        Fixed # of Identical Trials = n = 6
       P(x) = f(x)                                                P(x) = f(x)
   2   P(x = 2)                                                    0.098415        0.6
   2   P(x >= 2)                                                   0.114265        0.4
   2   P(x > 2)                                                     0.01585        0.2
   2   P(x < 2)                                                    0.885735          0
   2   P(x <= 2)                                                    0.98415                0   1   2


Mean   Σx*f(x)                                                           0.6
SD     sqrt(Σ(x-mean)^2*f(x))                                      0.734847
Mean   n*p                                                               0.6
SD     sqrt(n*p*(1-p))                                             0.734847

                                                                  X             f(x)
                                                                            0      0.531441
                                                                            1      0.354294
                                                                            2      0.098415
                                                                            3        0.01458
4   0.001215
5     5.4E-05
6   0.000001
3   4   5   6
n                5         Binomial Distribution, n = 5 p = 0.4
p              0.4
X                3


                                  Binomial Distribution, n = 5 p = 0.4
X       p(x)         0.4
    0     0.07776    0.3
    1       0.2592
                     0.2
    2       0.3456
    3       0.2304   0.1
    4       0.0768    0
    5     0.01024             0         1        2        3        4     5
n                        10                Binomial Distribution, n = 10 p = 0.5 P(X<=8)
p                        0.5               P(X<=8) = 0.9893
X                          8
Comparative
Operator           <=
                                                               Binomial Distribution, n = 10 p = 0.5 P(X<=8)
Result             <=8
BINOMDIST           0.989258 0.989258
                                                                                       0.246
                                                 P(X<=8) = 0.9893
X                  p(x)        P(X<=8)                                         0.205           0.205
               0    0.000977    0.000977
               1    0.009766    0.009766
               2    0.043945    0.043945                               0.117                           0.117
               3    0.117188    0.117188
               4    0.205078    0.205078
                                                               0.044                                           0.044
               5    0.246094    0.246094
                                               0.001   0.010
               6    0.205078    0.205078
               7    0.117188    0.117188
               8    0.043945    0.043945         0       1      2       3       4       5       6       7       8
               9    0.009766
              10    0.000977
0.5 P(X<=8)




      0.044
              0.010   0.001

               9       10
n                        10            Binomial Distribution, n = 10 p = 0.5 P(X<=4)
p                        0.5           P(X<=4) = 0.3770
X                          4
Comparative
Operator           <=
Result             <=4
BINOMDIST           0.376953

X                  p(x)      P(X<=4)
               0    0.000977
               1    0.009766
               2    0.043945
               3    0.117188
               4    0.205078
               5    0.246094
               6    0.205078
               7    0.117188
               8    0.043945
               9    0.009766
              10    0.000977
n                        10                Binomial Distribution, n = 10 p = 0.5 P(X<=4)
p                        0.5               P(X<=4) = 0.3770
X                          4
Comparative
Operator           <=                                         Binomial Distribution, n = 10 p = 0.5
Result             <=4
BINOMDIST           0.376953                                                P(X<=4)
                                                  0.3
X                  p(x)        P(X<=4)           0.25
               0    0.000977    0.000977          0.2
               1    0.009766    0.009766         0.15
               2    0.043945    0.043945          0.1
               3    0.117188    0.117188
                                                 0.05
               4    0.205078    0.205078
                                                    0
               5    0.246094
                                                          0      1      2      3      4    5   6   7
               6    0.205078
               7    0.117188                                                               x
               8    0.043945
               9    0.009766
              10    0.000977
= 10 p = 0.5




       8       9   10
    Poisson Probability Distribution can be used to estimate the number of occurrences over a
                                  specified interval of time or space
                                 Properties of a Poisson Experiment:
         The probability of an occurrence is the same for any two intervals of equal length
       The occurrence or nonoccurrence in any interval is independent of the occurrence or
                                 nonoccurrence in any other interval
    No upper limit for X: 0,1,2,3… , but as x increases past the mean, the probability decreases
                                         and gest quite small.
                                              Mean = Var
                                       Excel function: POISSON
                     POISSON(x,mean,cumulative (0 = exact, 1 = cumulative))


    x = Number of Web visitors                         7 per           minutes
    Business =                        Your Cool Web Site
           x = Number of Web visitors arrive at a rate of 7 per minute at Your Cool Web Site
    Assumption 1                            Probability is the same for any 1 minute interval
                                      Arrival or nonarrival of a web visitor in a 1 minute period is
                                         independent of the arrival or nonarrival in any other 1
    Assumption 2                                               minute period
    Mean                                               7 visits        per                        1 minute
a   x                                                  0 visits        per                        1 minute
    Mean
    P(0 visits per 1 minute)
    P(x) = (mu^x*e^-mu)/X!)
b   x                                                  2
    P(2 or more in 1 minute)
c   x                                                  1           0.5 minutes
    Mean
    P(1 or more in 30 sec.)
d   x                                                  5
    P(5 or more in 1 minute)




    x                                 p(x)                            x = Number of Web visitors arrive at a rate of 7 pe
                                    0    0.000911882                                           Cool Web Site
                                    1    0.006383174
                                                              0.16
                                    2    0.022341108
                                                              0.14
                    0.14
 3   0.052129252
                    0.12
 4   0.091226192     0.1
 5   0.127716668    0.08
 6    0.14900278    0.06
 7    0.14900278    0.04
 8   0.130377432    0.02
 9    0.10140467       0
10   0.070983269           0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
11   0.045171171                                   x = Number of Web visitors
12    0.02634985
13   0.014188381
14    0.00709419
15   0.003310622
16   0.001448397
17   0.000596399
18   0.000231933
19     8.5449E-05
20   2.99071E-05
21   9.96904E-06
22   3.17197E-06
23   9.65382E-07
24     2.8157E-07
25   7.88395E-08
26     2.1226E-08
27   5.50304E-09
28   1.37576E-09
29     3.3208E-10
30   7.74854E-11
rs arrive at a rate of 7 per minute at Your
 Cool Web Site
          15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
= Number of Web visitors
    Poisson Probability Distribution can be used to estimate the number of occurrences over a
                                  specified interval of time or space
                                 Properties of a Poisson Experiment:
         The probability of an occurrence is the same for any two intervals of equal length
       The occurrence or nonoccurrence in any interval is independent of the occurrence or
                                 nonoccurrence in any other interval
    No upper limit for X: 0,1,2,3… , but as x increases past the mean, the probability decreases
                                         and gest quite small.
                                              Mean = Var
                                       Excel function: POISSON
                     POISSON(x,mean,cumulative (0 = exact, 1 = cumulative))


    x = Number of Web visitors                         7 per           minutes
    Business =                        Your Cool Web Site
           x = Number of Web visitors arrive at a rate of 7 per minute at Your Cool Web Site
    Assumption 1                            Probability is the same for any 1 minute interval
                                      Arrival or nonarrival of a web visitor in a 1 minute period is
                                         independent of the arrival or nonarrival in any other 1
    Assumption 2                                               minute period
    Mean                                               7 visits        per                        1 minute
a   x                                                  0 visits        per                        1 minute
    Mean                                               7
    P(0 visits per 1 minute)              0.000911882                   =POISSON(C16,C17,0)
    P(x) = (mu^x*e^-mu)/X!)              0.0009118820                   =(C17^C16*EXP(1)^-C17)/FACT(C16)
b   x                                                  2
    P(2 or more in 1 minute)              0.992704944
c   x                                                  1           0.5 minutes
    Mean                                             3.5
    P(1 or more in 30 sec.)               0.969802617
d   x                                                  5
    P(5 or more in 1 minute)              0.827008392




    x                                p(x)                           x = Number of Web visitors arrive at a rate of 7 pe
                                   0    0.000911882                                          Cool Web Site
                                   1    0.006383174
                                                            0.16
                                   2    0.022341108
                                                            0.14
                    0.14
 3   0.052129252
                    0.12
 4   0.091226192     0.1
 5   0.127716668    0.08
 6    0.14900278    0.06
 7    0.14900278    0.04
 8   0.130377432    0.02
 9    0.10140467       0
10   0.070983269           0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
11   0.045171171                                   x = Number of Web visitors
12    0.02634985
13   0.014188381
14    0.00709419
15   0.003310622
16   0.001448397
17   0.000596399
18   0.000231933
19     8.5449E-05
20   2.99071E-05
21   9.96904E-06
22   3.17197E-06
23   9.65382E-07
24     2.8157E-07
25   7.88395E-08
26     2.1226E-08
27   5.50304E-09
28   1.37576E-09
29     3.3208E-10
30   7.74854E-11
rs arrive at a rate of 7 per minute at Your
 Cool Web Site
          15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
= Number of Web visitors

				
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Description: Success as an Insurance Agent document sample