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


by kingwinner
Tags: probability, px<1 or 2
kingwinner
kingwinner is offline
#1
Dec9-08, 08:22 AM
P: 1,270
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)=6x2y, 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
=∫ 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!
Phys.Org News Partner Science news on Phys.org
Simplicity is key to co-operative robots
Chemical vapor deposition used to grow atomic layer materials on top of each other
Earliest ancestor of land herbivores discovered
Avodyne
Avodyne is offline
#2
Dec9-08, 10:38 AM
Sci Advisor
P: 1,185
Your two formulas are the same at y=1, so it doesn't matter which one you use!
kingwinner
kingwinner is offline
#3
Dec10-08, 07:09 AM
P: 1,270
OK, but in general will they always be the same? What should we do in such a case in general?

Avodyne
Avodyne is offline
#4
Dec10-08, 11:02 AM
Sci Advisor
P: 1,185

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


If correctly derived from a given joint density function, yes, they must be the same.
kingwinner
kingwinner is offline
#5
Dec11-08, 12:06 AM
P: 1,270
um...Any proof about it?


Register to reply

Related Discussions
Joint probability from conditional probability? Set Theory, Logic, Probability, Statistics 10
What's the difference between probability and probability density Quantum Physics 5
How to find probability? Set Theory, Logic, Probability, Statistics 2
Probability / Probability density General Physics 2
Find the probability distribution Introductory Physics Homework 5