Statistical Distributions

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					 Statistical
Distributions
    Uniform Distribution
A R.V. is uniformly distributed on the
 interval (a,b) if it probability function
Fully defined by (a,b)

      P(x) = 1/(b-a)   for a <= x <= b
            = 0          otherwise
      Uniform Distribution
      Probability Function

  1




1/9



        1             10
Probability that x is between 2 and 7.5?
         Probability that x = 8?


  1




1/9



       1                       10
    Uniform Distribution
The cumulative distribution of a uniform
 RV is

    F(x) = 0           for x < a
         = (x-a)/(b-a) for a <= x <= b
          = 1          otherwise
    Uniform Distribution
    Cumulative Function
1




      1              10
  Uniform Distribution
 Discrete vs. Continuous
• Discrete RV
  – Number showing on a die
• Continuous RV
  – Time of arrival
  – When programming, make it discrete to
    some number of decimal places
     Uniform Distribution
• Mean = (a+b)/2
• Variance = (b-a)2 /12

• P (x < X < y) = F (y) – F (x)
  = (y-a) - (x-a) = y – x – a + a = y - x
     b-a     b-a        b–a         b-a
      Uniform - Example
A bus arrives at a bus stop every 20 minutes
  starting at 6:40 until 8:40. A passenger does
  not know the schedule but randomly arrives
  between 7:00 and 7:30 every morning. What
  is the probability the passenger waits more
  than 5 minutes.
       Uniform Solution
       X = RV, Uniform (0,30) -- i.e. 7:00 – 7:30
       Bus: 7:00, 7:20, 7:40
       Yellow Box <= 5 minute wait
   1




1/30

           A                 B        C

       5          10    15       20   25   30    40

               P (x > 5) = A + C = 1 – B = 5/6
     Arithmetic Mean
Given a set of measurements y1, y2,
 y3,… yn

Mean = (y1+y2+…yn) / n
           Variance
Variance of a set of measurements y1,
 y2, y3,… yn is the average of the
 deviations of the measurements
 about their mean (m).

  V = σ2 = (1/n) Σ (yi – m)2
                i=1..n
      Variance Example
Yi= 12, 10, 9, 8, 14, 7, 15, 6, 14, 10
m = 10.5
V= σ2 =   (1/10) ((12-10.5)2 + (10-10.5)2 +….
       = (1/10) (1.52 + .52 + 1.52….)
       = (1/10) (88.5)
       = 8.85
Standard Deviation = σ = 2.975
   Normal Distribution
• Has 2 parameters
  – Mean - μ
  – Variance – σ2
  – Also, Standard deviation - σ
Normal Dist.


                  .3413
                          .1359


                                  .0215
                                          .0013


-3   -2   -1              1   2      3
                 0
               Mean +- n σ
   Normal Distribution
• Standard Normal Distribution has
  – Mean = 0   StdDev = 1
• Convert non-standard to standard to
  use the tables
  Z value = # of StdDev from the mean
  Z is value used for reading table
          Z = (x – m)
                 σ
      Normal - Example
The scores on a college entrance exam are
 normally distributed with a mean of 75 and
 a standard deviation of 10. What % of
 scores fall between 70 & 90?

Z(70) = (70 – 75)/10 = - 0.5
Z(90) = (90 – 75)/10 = 1.5
.6915 - .5 = .1915 + .9332 - .5 = .4332
        = .6247 or 62.47%
 Exponential Distribution
A RV X is exponentially distributed with
  parameter  > 0 if probability function
Mean = 1/ 
Variance = 1 /  2   e = 2.71828182



   P(x) =
                       x
               e            For x >= 0

          =       0          Otherwise
Exponential Distribution
• Often used to model interarrival times
  when arrivals are random and those
  which are highly variable.
• In these instances lambda is a rate
  – e.g. Arrivals or services per hour
• Also models catastrophic component
     failure, e.g. light bulbs burning out
      Exponential Rates
• Engine fails every 3000 hours
  – Mean: Average lifetime is 3000 hours
  –  = 1/3000 = 0.00033333
• Arrivals are 5 every hour
  – Mean: Interarrival time is 12 minutes
  –  = 1 / 5 = 0.2


• Mean = 1 / 
Exponential Distribution
    Probability Function



            f(x)




                       x
        See handout for various graphs.
 Exponential Distribution
   Cumulative Function
Given Mean = 1/ 
Variance = 1/  2

F(x) = P (X <=x) = 1 – e -  x
Exponential Distribution
Cumulative Function (<=)


 1


            F(x)


             x
 Forgetfulness Property
Given: the occurrence of events conforms to
  an exponential distribution:
The probability of an event in the next x-
  unit time frame is independent on the time
  since the last event.
That is, the behavior during the next x-units
  of time is independent upon the behavior
  during the past y-units of time.
 Forgetfulness Example

• The lifetime of an electrical
  component is exponentially
  distributed with a mean of .
• What does this mean??
  Forgetfulness Examples
   The following all have the same probability
• Probability that a new component lasts the
  first 1000 hours.
• Probability that a component lasts the next
  1000 hours given that it has been working for
  2500 hours.
• Probability that a component lasts the next
  1000 hours given that I have no idea how long
  it has been working.
   Solution to Example
• Suppose the mean lifetime of
  the component is 3000 hours.
•  = 1/3000
• P(X >= 1000) = 1 – P(X <= 1000)
  1 – (1-e -1/3* 1) = e -1/3 = .717
How do we apply these?
1. We may be given the information
   that events occur according to a
   known distribution.
2. We may collect data and must
   determine if it conforms to a known
   distribution.

				
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