Solution to Homework 10

Problem 1

Let \(X_n\sim N(0, 1+1/n)\) and \(X\sim N(0,1)\). Find a joint distribution of \((X_n,X)\) so that \(X_n\rightarrow X\) in probability.

Problem 2

Let \(X\sim U[0,1]\) and \(X_n|X=x\sim U[x,x+1/n]\). Show that \(X_n\rightarrow X\) in probability.

Problem 3

Let \(X_n\) be a random variable with \(P(X_n=0)=1/n,P(X_n=1)=1-2/n,P(X_n=2)=1/n\). Show that \(X_n\rightarrow 1\) in probability

1. using the definition 3.2.6
   
2. using theorem 3.2.9a