Joint pdf and Conditional Expectation

[dondora]

hi there.

I'm solving probability problems.
And I've got some questions to ask to everybody here.
In a joint pdf of two continuous random variables X and Y,
I know how to calculate P[ 0 < X < 1 ] and P[ Y <= 1] and so on like
these things.
But I'm doing nothing to this problem P[X > Y].
Besides I'm not figuring out what it means and what I have to do.
Would you explain how to solve that and what it means?

Another thing, Given pdf of the continuous random variables X and Y,
I know E[ X | y ] = integral( x * fx(x | y) dx from -
infinity to +infinity
but I wonder what this E[X | Y] means. I scrutinized my text book but
it didn't cover it.
Do you know the difference between E[X | y] and E[X | Y] and how to
calculate E[ X | Y ]

Please help me. I'm going to wait for your clear-cut answer.
Thanks.

[Nasser Abbasi]

"dondora" <kon...@hanmail.net> wrote in message
news:1194883109.7...@y27g2000pre.googlegroups.com...

> hi there.
>
> I'm solving probability problems.
> And I've got some questions to ask to everybody here.
> In a joint pdf of two continuous random variables X and Y,
> I know how to calculate P[ 0 < X < 1 ] and P[ Y <= 1] and so on like
> these things.
> But I'm doing nothing to this problem P[X > Y].
> Besides I'm not figuring out what it means and what I have to do.
> Would you explain how to solve that and what it means?
>

P(X>Y) is the same as P(X-Y>0)

Hence, draw the line Y=X, and integrate the region below this line.

Hence P(X>Y)= double integral as follows (I think :)

x +inf
/ /
| | f(x,y) dx dy
/ /
-inf -inf

> Another thing, Given pdf of the continuous random variables X and Y,
> I know E[ X | y ] = integral( x * fx(x | y) dx from -
> infinity to +infinity
> but I wonder what this E[X | Y] means. I scrutinized my text book but
> it didn't cover it.

For a given value of r.v. Y, X has an expectation. So X is a function of y.
So we are talking about the expected value of a function of random variable.

> Do you know the difference between E[X | y] and E[X | Y] and how to
> calculate E[ X | Y ]
>

You just showed how to calculate E(X|Y)? it is the integral you wrote above.

As for the difference, the lower case letter is a specific value of the
random variable. the random variable is an UPPER CASE letter, and its value,
or realization, is lower case. the lower case y is NOT random, only upper
case letters are random.

One should really write E(X|Y=y) to be clear.

> Please help me. I'm going to wait for your clear-cut answer.
> Thanks.
>

Nasser