# 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)

cb34a135-5196-4a67-a768-b531ecd4b665.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