345 ACTIVITY 22: Exploring the Gamma and Exponential Distributions

WHY:

This activity introduces the Gamma function and distribution and shows how it is the parent of the exponential distribution which measures the time to the first occurrence and the time between occurrences in a Poisson process. Learning about exponential random variables helps complete our study of continuous random variables. Because of its ability to measure arrival times, the exponential distribution is very important when we try to simulate a system using a computer in order to predict its behavior and to forecast the consequences of important business decisions. Simulation is one of the most important applications of probability and statistics in business and industry.

LEARNING OBJECTIVES:

1. Discover how to apply the exponential random variable to a Poisson process.
2. Understand the relationship between the gamma and exponential distributions.
3. Understand the importance of the team captain in keeping the team on track and having fun critically thinking.

PERFORMANCE CRITERIA:

1. Quality of the answers to the Critical Thinking Questions.
   • Completeness
   • Clarity
   • Depth
2. Level of intensity reached by the team during the activity.

INFORMATION:

1. The gamma function gets its name from the symbol (the Greek letter gamma) used to represent it. It has wide application in math and especially in probability. It is defined as follows:

                      oo
              __      {
              | (a) = / xa-1 e-x dx
                      }
                      0
   					__ 	   __		 __
    By algebraic manipulation we see that  | (a+1) = a| (a) = a(a-1)| (a-1) ... 
        __			__
   and  | (1) = 1.  Thus,  | (n)  = (n-1)!. 
   

   This shows that the gamma function is a continuous extension to the factorial function.

   Another interesting property of the gamma function is its value when a = 1/2. It can be shown that

     	    __
               | (1/2) = \/pi.  
   

   For our purposes, the most important aspects of the gamma function are the probability distributions it generates.
   
   

2. The continuous random variable X has a gamma distribution, with parameters a > 0 and B > 0, if its probability density function is:

                   {  1
                   | ---__   xa-1 e-x/B   x > 0 
            f(x) = | Ba | (a)
   		|
                   {    0               elsewhere
   
        Its moment generating function can be computed as follows:
                oo
                 {     1
        MX(t) =  / etx ---__    xa-1 e-x/B
                 }    Ba | (a)
                 0
                         oo
                  1      {
              =  ---__   / xa-1 e-x(1-tB)/B dx
                 Ba | (a)}
                         0
   

   If we let y = x(1 - tB) then x^a-1 becomes (y/(1 - tB))^a-1 and dx becomes dy/(1 - tB). Substituting we get

   
                           oo
                     1      {  1
        MX(t) =  ---------  / ---__   ya-1 e-y/B dy
                 (1 - tB)a  } Ba | (a)
                            0
              =  (1 - tB)-a
   

   The last integral is 1 since it is the integral of the density function for a gamma random variable over its full range of values.

   Using the moment generating function we can verify that the expected value and variance of the gamma distribution are aB and aB² respectively.
   
   

3. If the random variable X measures the time to the first occurrence or the time between occurrences in a Poisson process then X has an exponential distribution and its probability density function is derived from the gamma density function by letting à = 1. Thus,

                   {  1
                   | --- e-x/B           x > 0 
            f(x) = |  B
                   {    0               elsewhere
   

   and E[X] = B and Var(X) = B². Note that the moment generating function for an exponential distribution is 1/(1 - tB).
   
   

4. The Poisson and exponential random variables are related in that the poisson random variable represents the number of occurrences in a time interval and the exponential random variable represents the time between occurrences. Their parameters _/\ and B are also related. Note that if the average time between occurrences is 20 minutes (1/3 hour) then the average number of occurrences in an hour is 3. This verifies that _/\ = 1/B.

RESOURCES:

1. Sections 6.5, 6.6: Probability and Statistics for E & S
2. Information Section on Activity Sheet
3. 30 minutes

PLAN:

1. Choose roles if you have not already done so.
2. Read the Activity and work through the Model.
3. Answer the Critical Thinking Questions

MODEL:

A Saint Mary's student is studying in her room and hoping that the phone will ring. She gets on the average 4 calls an hour. She would like to go down to the vending machines (a 10 minute trip) and asks you for the probability that she will miss a phone call if she leaves her room right after she hangs up on the previous call.

Let X be the time between phone calls. Since the occurrence of phone calls is a poisson process, X has an exponential distribution with B = 1/4 (i.e. the average time between calls is 15 minutes since she gets on the average 4 calls per hour. She doesn't talk very long when she gets a call). She wants to know the probability that the next phone call will come in the next 10 minutes = 1/6 hour. Thus,

                1/6               1/6
                 {  1              {                 1/6
P(X <= 1/6) =    / --- e-x/B dx  =  / 4 e-4x dx = -e-4x | = 1-e-2/3
                 }  B              }                  0
                 0                 0

CRITICAL THINKING QUESTIONS:

These will be provided in class.

SKILL EXERCISES:

1. Solve problem 8 on page 177.
   
   

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