Homework 1

Problem 1

Let \(X\) be a random variable with density \(f(x)=xe^{-x^2}\), \(x>0\). Find the variance of \(X\).

Problem 2

Let \(X\sim U[0,1]\) and \(Y|X=x\sim U[0,x]\), \(0<y<1\). Find the covariance of X and Y.

Problem 3

We have a sample from \(X_1,..,X_n\) are iid \(U[0,\theta]\):

0.005, 0.126, 0.582, 0.778, 1.109, 2.495, 2.610, 4.595, 7.926, 8.594

1. Find the mle of \(\theta\)

2. Test at the 5% level whether \(H_0:\theta=10\) vs \(H_a:\theta>10\) using a test based on the mle.