Find the distribution of W if its moment generating function is Mw(t) = (1/4 e^t + 3/4)^10 and calculate P(W ≤ 2). Use the following table. q = 1 - p.
pmf | Mean | Variance | mgf
Bernoulli X ~ Ber(p) | P(X = 1) = p, P(X = 0) = q | p | pq | pe^t + q
Binomial X ~ Bin(n, p) | P(X = k) = (nCk) p^k q^{n-k} for k = 0, 1, ..., n, where n is number of trials and k is the number of successes. | np | npq | (pe^t + q)^n
Geometric X ~ Geom(p) | P(X = n) = q^{n-1} p, where n is the number of trials. | 1/p | q/p^2 | pe^t / (1 - qe^t)
Negative Binomial X ~ Negbin(k, p) | P(X = n) = (n-1Ck-1) p^k q^{n-k} for n ≥ k. where n is number of trials and k is the number of successes. | k/p | kq/p^2 | (pe^t / (1 - qe^t))^k
Poisson X ~ Poisson(λ) | P(X = k) = e^{-λ} λ^k / k! where k is the number of successes that occur in a specified region. | λ | λ | e^{λ(e^t - 1)}
Hypergeometric X ~ Hypergeom(N, NA, n) | P(X = k) = (NA C k)(N-NA C n-k) / (N C n) for k = 0, 1, ..., n. where NA is the number of type A in the pocket, k is the number of items of type A selected, N is the number of total items in the pocket, and n is the number of total items selected. | nNA/N | (N-n)/(N-1) n NA(N-NA)/N^2 | *