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270 Chapter 4 Special Distributions 4.5.9. Differentiate the moment-generating function 1 0 pet 1â(1â p)et r M X (t) = to verify the formula given in Theorem 4.5.1 for E(X ). 4.5.10. Suppose that X 1 , X 2 , . . . , X k are independent negative binomial random variables with parameters r1 and p, r2 and p, . . ., and rk and p, respectively. Let X = X 1 + X 2 + Â· Â· Â· + X k . Find M X (t), p X (t), E(X ), and Var(X ). 4.6 The Gamma Distribution Suppose a series of independent events are occurring at the constant rate of Î» per unit time. If the random variable Y denotes the interval between consecutive occurrences, we know from Theorem 4.2.3 that f Y (y) = Î»eâÎ»y , y > 0. Equivalently, Y can be interpreted as the âwaiting timeâ for the ï¬rst occurrence. This section generalizes the Poisson/exponential relationship and focuses on the interval, or waiting time, required for the rth event to occur (see Figure 4.6.1). Figure 4.6.1 Y Time 0 Theorem 4.6.1 First success Second success rth success Suppose that Poisson events are occurring at the constant rate of Î» per unit time. Let the random variable Y denote the waiting time for the rth event. Then Y has pdf f Y (y), where Î»r y r â1 eâÎ»y , y > 0 f Y (y) = (r â 1)! Proof We will establish the formula for f Y (y) by deriving and differentiating its cdf, FY (y). Let Y denote the waiting time to the rth occurrence. Then FY (y) = P(Y â¤ y) = 1 â P(Y > y) = 1 â P(Fewer than r events occur in [0, y]) =1â r â1 eâÎ»y k=0 (Î»y)k k! since the number of events that occur in the interval [0, y] is a Poisson random variable with parameter Î»y. From Theorem 3.4.1, ( ) r â1 k d âÎ»y (Î»y) e f Y (y) = FY (y) = 1â dy k! k=0 = r â1 (Î»y)k âÎ»y (Î»y)kâ1 â Î»e k! (k â 1)! k=1 Î»eâÎ»y (Î»y)k âÎ»y (Î»y)k â Î»e k! k! k=0 k=0 = r â1 k=0 = r â1 Î»eâÎ»y Î»r y r â1 eâÎ»y , (r â 1)! r â2 y >0