STAT 801

Problems: Assignment 2

Postscript version of these questions

1.

Suppose X and Y have joint density f(x,y). Prove from the definition of density that the density of X is $g(x)=\int f(x,y)\, dy$.

2.

Suppose X is Poisson($\theta$). After observing X a coin landing Heads with probability p is tossed X times. Let Y be the number of Heads and Z be the number of Tails. Find the joint and marginal distributions of Y and Z.

3.

Let p₁ be the bivariate normal density with mean 0, unit variances and correlation $\rho$ and let p₂ be the standard bivariate normal density. Let p= (p₁+p₂)/2.

(a)

Show that p has normal margins but is not bivariate normal.

(b)

Generalize the construction to show that there rv's X and Y such that X and Y are each standard normal, X and Y are uncorrelated but X and Y are not independent.

4.

Warning: This is probably hard. Don't waste too much time on it. Suppose X and Y are independent $\Gamma(p,1)$ and $\Gamma(p+1/2,1)$ random variables. Show that Z=2(XY)^1/2 is a $\Gamma(2p,1)$ random variable.

5.

Suppose X and Y are independent with $X \sim N( \mu, \sigma^2 )$ and $Y \sim N( \gamma , \tau^2 )$. Let Z=X+Y. Find the distribution of Z given X and that of X given Z.