For this exercise, let X denote the number of failures that occur before the r-th success....

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For this exercise, let X denote the number of failures that occur before the r-th success. Then, we can write the probability mass function for the Negative Binomial Distribution as P(X = k) = ¹)p'd', (a) Verify that (b) Show that E[X]: = r+k-1 k Hint: Use the 'negative' binomial series (1 - x) rq Р Σ P(x = k) = 1. P(X k=0 and Var(X) where k = 0, 1, 2, .... = rq = ·Σ (r + k − ¹) 2². k=0 For this exercise, let X denote the number of failures that occur before the r-th success. Then, we can write the probability mass function for the Negative Binomial Distribution as P(X = k) = ¹)p'd', (a) Verify that (b) Show that E[X]: = r+k-1 k Hint: Use the 'negative' binomial series (1 - x) rq Р Σ P(x = k) = 1. P(X k=0 and Var(X) where k = 0, 1, 2, .... = rq = ·Σ (r + k − ¹) 2². k=0

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