Homework 6

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.