Joint Probability Density Function

[twoski]

Homework Statement

The joint probability density function of X and Y is given by
f(x, y) = c( x³ + xy/4 )

0 < x < 1
0 < y < 2

(a) For what value of c is this a joint density function?
(b) Using this value of c, compute the density function of Y .
(c) Using this value of c, nd PfX > Y g.

The Attempt at a Solution

I'm looking through my notes and i can't find anything that helps me solve A... :(

 

[LCKurtz]

The double integral, over the region, of the density function must be 1.

 

[twoski]

So the idea is, if i do a double integral on this function i should end up with c * some value, and i need to find a value for c such that c * some value = 1?

 

[Ray Vickson]

So the idea is, if i do a double integral on this function i should end up with c * some value, and i need to find a value for c such that c * some value = 1?

Try it yourself to see!

 

[twoski]

So i did it and i ended up with 3/4 * c... So the idea now is that i want C to equal 1?

And how would i start computing the density function of Y?

 

[Ray Vickson]

So i did it and i ended up with 3/4 * c... So the idea now is that i want C to equal 1?

And how would i start computing the density function of Y?

Do you not understand that you want the double integral = 1? That means that you need to equate your result to 1.

As to the density of Y: it will just be the *marginal* density of Y. The meaning of this was already dealt with in another thread.

 

[twoski]

Okay I've managed to figure out everything except for the last question. I have formulas for P( X ≤ x ) but it's asking me to find P( X < Y ).

 

[LCKurtz]

Okay I've managed to figure out everything except for the last question. I have formulas for P( X ≤ x ) but it's asking me to find P( X < Y ).

Draw a picture of that region inside your given domain and integrate the joint density over it.