Homework 6

Do as much as you can analytically, and for the rest use R.

Problem 1

A rv X has a discrete uniform distribution on 1,..,N if P(X=k)=1/N for \(1\le k\le N\). Here N is some fixed positive integer. We write \(X\sim U\{1,..,N\}\).

1. Say \(X\sim U\{1,..,N\}\). Find E[X] and Var[X].

2. Say \(X_1,..,X_k\sim U\{1,..,N\}\) and independent. Say \(M_k=\max\{X_1,..,X_k\}\). Find \(E[M_k]\).

Data from a discrete uniform can be generated with the R sample command. Use this to verify your answers for N=10 and k=2.

Problem 2

Say \(X\sim Exp(1)\) and \(Y|X=x \sim Pois(x)\). Find Var[Y].

Problem 3

Say \(X\sim N(0, 1)\).

1. Find the moment generating function of X, defined by

\[\psi(t)=E[e^{tX}]\]

2. Show that \(P(X>a)\le e^{t^2/2-ta}\) for any t>0 and a>0

3. Show that \(P(X>a)\le e^{-a^2/2}\) for any a>0.