I am working on a problem and am a bit confused.
The problem: Consider (X,Y) with density p(x,y) = 24xy on
0$\le$x$\le$1
0$\le$y$\le$1
0$\le$x+y$\le$1
Find: P(X$\le$.5 | Y$\ge$.5)
What I have done so far:
P(X$\le$.5 | Y$\ge$.5) = $P(X\le.5) \over P(Y\ge.5)$
I imagine the setup might look something along the lines of:
P(X$\le.5$) = 1 - $\int_{.5}^1$ $\int_{.5}^1$ 24xy d(x) d(y)
I am unsure how to determine the integral set ups however.
As BGM has said you have been asked to find the conditional probability, i.e. given Y > 0.5 what is the probability X < 0.5. However, there is no indication that the two variables are in any way dependant on each other. Maybe they are? This would make sense but necessitate an approach that we need more information for.
It is more likely you have been asked for the joint probability, i.e. P(X<0.5,Y>0.5) no vertical bar. In this case, yes, you want to integrate 24xy over the specified limits, but wait a minute does an integration of the density over the entire space = 1? It turns out it doesn't. You will have to normalise it by dividing by this value.
A final note is that in mathematics case matters X does not necessarily = x. Has something been left out or mistyped?