- #1
kingwinner
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Homework Statement
Suppose X and Y are jointly continuous random variables with joint density function
f(x,y)=6x2y, 0<x<y, x+y<2
f(x,y)=0, otherwise
Find P(X<1/2 | Y=1).
Homework Equations
The Attempt at a Solution
By definition,
P(X<1/2 | Y=1)
1/2
=∫ fX|Y(x|y=1) dx
-∞
My computations:
Marginal density of Y:
fY(y)=2y^4, 0<y<1
fY(y)=2y(2-y)^3, 1<y<2
Condition density of X given Y=y:
Case 1: For given/fixed 0<y<1,
fX|Y(x|y)=3x^2 / y^3, 0<x<y
Case 2: For given/fixed 1<y<2,
fX|Y(x|y)=3x^2 / (2-y)^3, 0<x<2-y
I hope these are correct. Now P(X<1/2 | Y=1) is the troublesome case because we are given Y=1, which formula for fX|Y(x|y) should I use?
Thanks for any help!