- #1

tomelwood

- 34

- 0

## Homework Statement

Hi, at the moment I am trying to revise for my Probability exam, and a couple of the questions on the past paper are as follows, however I can find nothing in our notes that is of any use! Any help would be greatly appreciated, thankyou.

i) Two random variables X and Y have density function:

f[tex]_{X,Y}[/tex](x,y) = axy[tex]^{2}[/tex] if 0[tex]\leq[/tex]x[tex]\leq[/tex]1, 0[tex]\leq[/tex]y[tex]\leq[/tex]1 and 0 otherwise

Determine the constant a such that f is a density function.

ii)Let (X,Y) be a random vector with density function:

f[tex]_{X,Y}[/tex](x,y) = 6x , if 0<x<y<1 , 0 otherwise

Compute Cov(X,Y) and [tex]\rho[/tex](X,Y) , where [tex]\rho[/tex] is the correlation.

## Homework Equations

## The Attempt at a Solution

i) For this one I'm not entirely sure what to do here. I feel I should integrate it to give me the pdf, F, but then I don't know what to do with it. What condition should it satisfy that I can impose to give me a set value for 'a'?

ii)For this, I know that the formula for Covariance is E[(X-E[X])(Y-E[Y])] and that the correlation is [tex]\frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}}[/tex]

So does that mean that Cov(X,Y) is just E[g(X,Y)] with g(X,Y) equalling X and Y respectively?

In which case E[X] is just the integral of the density function multiplied by x (or y) as in the 1 dimensional case?

(You may have noticed I had a moment of inspiration halfway through writing this, but have carried on as I am not sure if my inspiration is correct!)

Many thanks in advance.