Suppose has the bivariate normal distribution with density function
Show that
First you should recognise from the joint pdf that and .
Therefore . Mr F Edit: The mistake I mention below is here. See post #3.
Note: .
So the task is to calculate :
Now complete the square and re-arrange:
where is the pdf of .
Now note that represents and so:
.
Therefore:
where the moments are for a standard normal distribution. Cheating and refering to a table of moments (see Normal distribution - Wikipedia, the free encyclopedia):
.
Therefore .
This is different from the result that was given to be proved so there's a mistake somewhere (that I don't have time to look for). Nevertheless, I'm sure you get the idea.