Conditional Density question!

Throughout this question

*X *and *Y *will be continuous random

variables which have joint probability density function

f

X;Y

( *x, y*) = *c x y*2 if 0 *< y < x < *4 *;*

* = 0 otherwise*

where *c *is some non-negative constant. Do the following:

(a) Determine the constant *c*. Answer =

15/

45 *:*

(b) Calculate the marginal density function

*fY *of *Y *. Answer =

15

2 (45) [ 16 *y*2 -*y*4 ] *:*

(c) Calculate

P [ 4 *< X *+ *Y *] . Answer =

59/

64

:

(d) Calculate

P [ *Y < *2 *j *4 *< X *+ *Y *] . Answer =

12/

59

:

(e) Calculate

P [ 2 *< X j Y *= 1 ] . Answer =

4/

5

Hey guys, this is a question on one of my tutorial sheets, Ive worked through the first 4 sucessfully but cant seem to get the correct answer for (e). We were given answers in class annd told to workout ourselves. Im just not sure am I doing it correctly. Here are the steps I took.

1: I drew the region on a graph, the x- values ran from x = 2 to x = 4 (for the purpose of integration limits) and the y values ran from y = 1 to y = x.

2: I found the joint density which I got to be 5772/6(45)

*3: I worked out the value of the marginal density of y at y=1. And then divided this number into the joint density to get the conditional density. *

Im just wondering where I have gone wrong and if anyone could give a helping hand

Thanks so much

Nappy