The geometric distribution concerns the trial, x, at which the first success is observed in a sequence of iid bernoulli trials with parameter, p. the negative binomial distribution concerns the trial, y, at which success r is observed in a sequence of iid bernoulli trials with parameter, p. the pmf of the negative binomialar, p) distribution is: f(y) = (% = 1)p"(1 – p)»-", y=r, r +
(a) explain in words with minimal math) why a negative binomial random variable can be viewed as the sum of riid geometric random variables with parameter p: y = x1 ++x, (2 pts]
(b) derive the mgf of y as pet for t < -log(1 - p). (1 - (1 - pet] (4 pts)
(c) derive the mean and variance of y as e[y] = and var(y) = " . (hint: use mgf.) 2 pts]
(d) a gambler plans to bet on red in the roulette game of red-and-black. the roulette wheel has 18 red slots, 18 black slots, and 2 green slots, all equally likely. how many games does the gambler expect to play before she wins 5 times? what is the probability that the gambler plays at least 15 times before she gets her fifth win?