Discrete Random Variables (cont.) Discrete Distributions the Geometric pmf

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1 Discrete Random Variables (cont.) ECE 313 Probability with Engineering Applications Lecture 10 - September 29, 1999 Professor Ravi K. Iyer University of Illinois the Geometric pmf Consider a sequence of Bernoulli trials, where we count the number of trials until the first ÒsuccessÓ occurs. Let 0 denote a failure and let 1 denote a success, then the sample space consists of the set of all binary strings with an arbitrary number of 0 s followed by a single 1. S {0 i-1 1 i 1, 2, 3,...} (S is a countably infinite set) Define a random variable Z on this sample space so that the value assigned to the sample point 0 i-1 1 is i. Thus Z is a random variable with image {1, 2,...}, which is a countably infinite set

2 To find the pmf of Z note that the event [Z i] occurs if and only if we have a sequence of i - 1failures followed by one success. This is a sequence of independent Bernoulli trials with the probability of success equal to p. Hence, we have for i 1, 2,..., (A) p Z (i) q p p(1 p) where q 1 - p. By the formula for the sum of a geometric series, we have: p () i pq Z i 1 i 1 1 p p q p 1 Any random variable with the image {1, 2,...} and pmf given by the formula of the form of equation (A) is said to have a geometric distribution and the function given by (A) is called a geometric pmf with parameter p. The corresponding CDF is: F (t) p(1 p) 1 (1 p) Z t i 1 t for t 0

3 the Modified Geometric pmf (cont.) The random variable Z (defined on the previous slides) counts the total number of trials up to and including the first success. We are often interested in counting the number f failures before the first success. Let this number be called the random variable X with the image {0, 1, 2,...}. Clearly, Z X + 1. the Modified Geometric pmf (cont.) The random variable X is said to have a modified geometric pmf, specify by p X (i) p(1 p) The corresponding distribution function is: i for i 1, 2,..., i F ( t) p( 1 p) 1 ( 1 p) X t i 0 t+ 1 for t 0

4 Examples where the geometric distribution occurs include 1. A series of components made by a certain manufacturer. The probability the ith item is defective one is given by the geometric pmf. 2. Consider the operation of a time-sharing computer system with a fixed time-slice. The pmf of the random variable denoting the number of time slices needed to complete the execution of a program is given by geometric pmf. 3. Consider the program segment consisting of a while loop: while  B do S If the successive test of the Boolean expression B are independent, then the number of times the body (or the statement group S) of the loop is executed will be a random variable having a modified geometric distribution with parameter p (probability the B is true) 4. Consider a repeat loop repeat S until B The number of times the body of the repeat loop is executed will be a geometrically distributed random variable with parameter p.

5 Memoryless Markov Property Consider a sequence of Bernoulli trials and let Z represent the number of trials until the first success. Assume that we have observed a fixed number n of these trials and found them all to be failures. Let Y denote the number of additional trials that must be performed until the first success. Memoryless Markov Property (cont.) Then Y Z - n, and the conditional probability q i is: qi P( Y i Z > n) PZ ( n i Z> n) PZ ( n+ i Z> n) PZ ( n+ iand Z> n) PZ ( > n) by using the definition of conditional probability.

6 Memoryless Markov Property (cont.) For i 1,2,3,É., Z n + i implies that Z > n. Thus the event [Z n + i and Z > n] is the same as the event [Z n + i]. Therefore: qi P( Y i Z > n) PZ ( n+ i) PZ ( > n) pz ( n+ i) 1 FZ ( n) n+ i 1 pq n 1 ( 1 q ) n+ i 1 pq n q pq p () i z Conditioned on Z > n, the number of trials remaining until the first success, Y Z - n, has the same pmf as Z had originally. If a run of failures is observed in a sequence of Bernoulli trials, we need not remember how long the run was to determine the probabilities for the number of additional trials needed until the first success. The geometric distribution is the only discrete distribution with Markov property. Negative Binomial pmf Let us observe the number of trials until the rth success, and let T r be the random variable denoting this number. The image of T r is {r, r+1, r+2,...} To compute p T (n) define the events: r A ÒTr n.ó B ÒExactly r - 1 successes occur in n -1 trials.ó C ÒThe nth trial results in a success.ó Then A B C and the events B and C are independent. Therefore: P(A) P(B)P(C).

7 Negative Binomial pmf (cont.) To compute P(B), consider a particular sequence of n - 1 trials with r - 1 successes and n (r - 1) n - r failures. The probability associated with such a sequence is p r-1 q n-r and there n 1 are such sequences. r 1 n r n r Therefore: 1 1 PB ( ) p r 1 q Since P(C) p, p ( n ) T P ( T r n ) r PA ( ) n 1 r pq r 1 n r n 1 r n r p ( 1 p), n r, r+ 1, r+ 2,... r 1