By Van Der Merwe A. J., Du Plessis J. L.

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X denote the trial number on which this first success occurs. Then the pmf of X is given by because there must be x - 1 failures before the first success occurs on trial x. v. X defined by Eq. v. with parameter p. (a) Show that px(x) given by Eq. 67) satisfies Eq. 17). (b) Find the cdf F,(x) of X. (a) Recall that for a geometric series, the sum is given by Thus, (b) Using Eq. v. X of Prob. v. with p == 4. 16. v. with parameters (n, p). (a) Show that p&) given by Eq. 36) satisfies Eq. 17). RANDOM VARIABLES [CHAP 2 (b) FindP(X> l ) i f n = 6 a n d p = 0 .

2-7 Uniform distribution over (a, b). v. X is often used where we have no prior knowledge of the actual pdf and all continuous values in some range seem equally likely (Prob. 69). E. v. v. with parameter A (>O) if its pdf is given by which is sketched in Fig. 2-8(a). The corresponding cdf of X is which is sketched in Fig. 2-8(b). Fig. 2-8 Exponential distribution. v. X are (Prob. 32) The most interesting property of the exponential distribution is its "memoryless" property. By this we mean that if the lifetime of an item is exponentially distributed, then an item which has been in use for some hours is as good as a new item with regard to the amount of time remaining until the item fails.

C. v. v. with parameter A (>0) if its pmf is given by The corresponding cdf of X is Figure 2-6 illustrates the Poisson distribution for A = 3. CHAP. 21 RANDOM VARIABLES Fig. 2-6 Poisson distribution with A = 3. v. X are (Prob. 29) px = E(X) = A. v. v. with parameters (n, p ) when n is large and p is small enough so that np is of a moderate size (Prob. 40). 's include 1. The number of telephone calls arriving at a switching center during various intervals of time 2. The number of misprints on a page of a book 3.