STAT 230 Final, Spring 1998

\ Mgyh 233 Spring 1998 Final Exam Ve . Let X be ar.v. with the following p.df. f(x)=1/4 -l<x<0 W2 x=0 7/4 O<x<l 0 elsewhere What is the d.f. of X, F(x) ? 2. Let X be ar.v. with the following d.f. F(x)= 0 O<x ¥/2 O<x«<l /2 isx<2 1 2x What is the p.d.f. of X, f(x) ? 3. Let X and Y have the following joint p.df.; f(x,y) = xye™ O<y<x, 0<x<m 3 elsewhere a). What is the marginal p.d.f. of X, ie. f; (x) ? b). What is the marginal p.d.f. of Y, ie. f; (y) ? c). What is the ECYIX) ? d). What is the E(Y) ? 4. Let X and Y be independent r.v. each with an N(u, o” ) distribution. a). What is the distribution of V=X+Y? b). What is the distribution of W =X - Y ? c). Assume that V and W have a bivariate normal distribution. Are V and W independent? 5. Let Xi, X2,..., X_ be mutually independent r.v.’s, each with a (1, 1) distribution. a). Identify the distribution of Z=X, +X. +... +X,. Let Y = max(X), Xz, ..., Xn). b). What is the df. of Y? c). What is the p.d.f. of Y? 6. Acat chases a mouse through the streets of a city. The cat can only catch the mouse, with probability 1/2, when the mouse stops to cross a street. The chase continues until the cat catches the mouse. a). What is the probability that the cat catches the mouse on the n™ street crossing, n=1,2,3,...? b). What is the probability that the cat catches the mouse? c). What is the expected number of street crossings in the chase? 7. Let X ~N(u, 0”). Whatis the distribution of Y = (X — Ly? ? Show all work. 8. Let X ~ b(n, p), where n is a realization or experimental value of the r.v. N which has a Poisson distribution with parameter A. a). What is the ECXIN=n) ? b). What is the var(XIN=n) ? c). What is the E(X) ? d). What is the var(X) ?