Question

Problems binomial random variable has the moment generating function ψ(t)-E( ur,+1-P)". Show, that EIX) np and Var(X)-np(1-P) using that EXI-v(0) and Elr_ 2. Lex X be uniformly distributed over (a b). Show that EX]- and Varm-ftT using the first and second moments of this random variable where the pdf of X is () Note that the nth i of a continuous random variable is defined as E (X%二z"f(z)dz. (z-p?expl- ]dr. ơ, Hint./ udv-w-frdu and r.e-//agu-VE. 3. Show that 4 The moment generating function of the normal distribution with parameters μ and σ2 is ψ(t)- exp (pat +扣2,2) for -oo < t < oo. Show that Els-e(0)-μ and Var(X)-r(0)-W(0)12 = 5. Suppose that Xi.X, and X are independent random variables such that Ex.]-0 and ELX?] -1 for i12,3. Find the value of EXj(2X1-X3)1 6 Suppose that X and Y are random variables such that Var(x)-Var(V)-2 and Cov(x,Y)-1. Find the value of Var(3X-Y+2) 7. Show that ECx)-O and Var ifXu . .. , x, are independent and identically distributed with E(X)-0 and E(X?)-o? for-1, ,

Answer 1