(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose X and Y are jointly continuous random variables with joint density function

f(x,y)=6x^{2}y, 0<x<y, x+y<2

f(x,y)=0, otherwise

Find P(X<1/2 | Y=1).

2. Relevant equations

3. The attempt at a solution

By definition,

P(X<1/2 | Y=1)

1/2

=∫ f_{X|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,

f_{X|Y}(x|y)=3x^2 / y^3, 0<x<y

Case 2: For given/fixed 1<y<2,

f_{X|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 f_{X|Y}(x|y) should I use?

Thanks for any help!

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# Find the Probability: P(X<1/2 | Y=1)

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