Problem Involving Calculating the probability given the marginal density functions!
I'm trying to solve this problem:
If X and Y are independent and f(x) = 1/2 when 0<X<2 and f(y) = 2Y when 0<Y<1, find P(X<2Y).
As X and Y are independent, f(x,y) = Y when 0<X<2 and 0<Y<1.
I integrated Y first with respect to X from 0 to 2Y, then I integrated this with respect to Y from 0 to 1, which gives 2/3 for the answer.
Am I right?
Re: Problem Involving Calculating the probability given the marginal density function
That approach is spot on, but you need to consider X/2 < Y in which the order of integration is 0 to Y/2 for X and 0 to 1 for Y.
Changing the order of integration gives 0 to 1/2 for X and 0 to 1 - 2X for Y.
Integrating this with respect to dy with limits (0 to 1-2x) first gives (1/2)*(1-2X)^2 and integrating this over dx with limits (0 to 1/2) gives 1/6.
Double check my calculations for the integration limits, but the approach you had to setup the PDF is right (and if I'm wrong about the integration please inform).