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Let $f_{X,Y}(x,y) = kxy$, for $0 ≤ x, y ≤ 1,$ otherwise $f_{X,Y}(x,y) = 0$.

(a) Determine $k$ such that $f_{X,Y}(x,y)$ is a PDF.

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Reminder: A property of PDF is such that its integral over its domain equals what. Apply your calculus. I think you know what the what is. – Frenzy Li Aug 12 '12 at 12:26
up vote 2 down vote accepted

Selected from PDF at WikiPedia.

The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one.

As implied, $\int_{R^2} f_{X,Y}(x,y)=1$. (Property of PDF)

Thus $\int_0^1\int_0^1 kxy dxdy=1$, with the rest being mere integration.

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k = 4? :) Thanks so much! – Fred Aug 12 '12 at 12:37
@Panda Yep, that's the answer. Harder questions involve triangular or circular area of integration which will just complicate integration, though-- e.g. $0<=x<=y<=1$. – Frenzy Li Aug 12 '12 at 12:43
@Panda Welcome, everyboy gets to learn. – Frenzy Li Aug 12 '12 at 23:49

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