Continuous joint random variables Watch

Josh1994yetman
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This question has me stumped. I hope someone can at least point me in the right direction:

QUESTION: Two continuous random variables X and Y have joint probability density function given by:

fX,Y(x,y)= x(x+cy) 0<x<1, 0<y<2, 0 otherwise;

a) Find the value of c (I got c=1/3 for this part).

b) Find the marginal density of X and find P(X>1/2) (I got 2x^2+(2/3)x for the marginal density, and P(X>1/2)=5/6).

c) Find P( Y<1/2 | X < 1/2) (dunno about this one)
d) Find P(Y<X) (or this one)

Any help is appreciated
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DFranklin
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P(Y < 1/2 | X < 1/2) = P(Y<1/2 and X<1/2) / P(X<1/2).

P(Y &lt; \frac{1}{2} \cap X &lt; \frac{1}{2}) = \int_{x=0}^{1/2} \int_{y=0}^{1/2} f(x, y) \,dy \,dx

For P(Y < X) work out what the region where Y<X is and integrate f(x, y) over that region.
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Josh1994yetman
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How does one identify this region? Thanks for your quick response earlier
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ghostwalker
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(Original post by Josh1994yetman)
How does one identify this region? Thanks for your quick response earlier
Draw the graph Y=X, with appropriate limits and shade the desired region. Et voila.
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