Exercise 1 LetX and Y be two independent Poisson random variables with distribution parameters A1 and A2, respectively. Show that the conditional distribution of X given X + Y is binomial. (Hint: Note thatP(X = m|X + Y = n) = W) Exercise 2 Suppose that the life of a certain light bulb is exponentially distributed with mean 100 hours. If 10 such light bulbs are installed simultaneously, (a) What is the distribution of the life of the light bulb that fails last? (b) What is the distribution of the life of the light bulb that fails second? Exercise 5 Let X1- has a mean ,u and variance of, and X2- and X j are (pain/vise) inde- pendent forz', j = 1, ...,n. Let Sn 2 227:1 (1in- for some constants a1, ...,a,n such that 22:1 a2- = 1. Get (an, ..., an) that minimizes Var(Sn). Exercise 7 Let (X1, X2, .., X\") be an independent and identically distributed random sample from a cdf FX(-). Define FX(:I:) = i2; 1(X2- g x) be an empirical cdf of random variable X. Show that for each a: (a) nx) ~ Bin(n, FX(m)), a binomial distribution with (n, FX(x)). (b) EiFX$)] = FXW- (C) Var[FX(33)] = n'lFX(x)(1 FX($))