# Question: The graphs of the moment generating functions of three normal distributions N 0

The graphs of the moment-generating functions of three normal distributions—N(0, 1), N(−1, 1), and N(2, 1)—are given in Figure 3.3-3(a). Identify them.

## Answer to relevant Questions

If Z is N(0, 1), find (a) P(0 ≤ Z ≤ 0.87). (b) P(−2.64 ≤ Z ≤ 0). (c) P(−2.13 ≤ Z ≤ −0.56). (d) P(|Z| > 1.39). (e) P(Z < −1.62). (f) P(|Z| > 1). (g) P(|Z| > 2). (h) P(|Z| > 3). Let X have an exponential distribution with θ = 1; that is, the pdf of X is f(x) = e−x, 0 < x < ∞. Let T be defined by T = ln X, so that the cdf of T is G(t) = P(ln X ≤ t) = P(X ≤ et) (a) Show that the pdf of T is ...Determine the indicated probabilities from the graph of the second cdf of X in Figure 3.4-4: (a) P (−1/2 ≤ X ≤ ½) . (b) P (1/2 < X< 1) . (c) P (3/4 < X < 2). (d) P(X > 1). (e) P(2 < X < 3). (f) P(2 < X ≤ 3). Let X and Y have a trinomial distribution with parameters n = 3, pX = 1/6, and pY = 1/2. Find (a) E(X). (b) E(Y). (c) Var(X). (d) Var(Y). (e) Cov(X, Y). (f) ρ. Show that in the bivariate situation, E is a linear or distributive operator. That is, for constants a1 and a2, show thatPost your question