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$X$,$Y$ are independent random variables, whose density function is $f(x,y)$.

To get the Probability of $X, I use the integration of the area $[-\infty,y]\times[-\infty,+\infty]$.

$P(X

$f(x,y)=f_{X}(x)f_{Y}(y)$

$\int_{-\infty}^ydx\int_{-\infty}^{+\infty}f(x,y)dy=\int_{-\infty}^ydx\int_{-\infty}^{+\infty}f_{X}(x)f_{Y}(y)dy=\int_{-\infty}^y[f_{X}(x)\int_{-\infty}^{+\infty}f_{Y}(y)dy]dx=\int_{-\infty}^y[f_{X}(x)\int_{-\infty}^{+\infty}f_{Y}(y)dy]dx=\int_{-\infty}^yf_{X}(x)dx=F_X(y)$

But we know that

$P(X

What's wrong with my calculation?

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    We want $P(X, so $x$ can only go up to $y$. Despite too many years integrating, I still *always* sketch the region.2012-05-09

1 Answers 1