Statistics

2. Suppose that Y follows a Gamma(a, A) distribution. (a) (2 pts) When c is any positive or negative value such that a + c > 0, find E(Yc) using the integration by pattern recognition. (b) (2 pts) If X follows a x? distribution, what is E(Xc) for a constant c defined in (a)? 3. Let X ~ Pois()). (a) (3 pts) Find the MGF of X. Hint: et = Zin=0 n!. (b) (2 pts) Use the MGF to show that E(X) = 1. Hint: deg(t) = g'(t)es(t). (c) (2 pts) Use the MGF to show that Var(X) = A. Hint: Ag(t)h(t) = g'(t)h(t) + g(t)h'(t). 4. Let X ~ Geom(p). The MGF was derived during the class. (a) (2 pts) Using the MGF, show that E(X) = 1/p. Hint: d 9(t) g' (t) h (t) - g(t ) h' (t) dt h(t) h (t ) 2 (b) (2 pts) Using the MGF, show that Var(X) = (1 -p) /p2. 5. (3 pts) Find the MGF of Y ~ NBin(r, p)