- #1
obesogen
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Homework Statement
If X,Y are independent RVs ~ U[-1,1], and W=X*Y, find the density of W.
Homework Equations
The Attempt at a Solution
I feel my approach is right, but the bounds of my final integral don't make sense.
[tex]
F_{XY}(w)=P(XY \leq w)=\int_{-\infty}^{\infty} P(XY \leq w | Y=y)*f_Y(y)dy \\
=\int_{-\infty}^{\infty}P(X*y \leq w)*f_Y(y)dy \\
=\int_{-1}^{1}P(X*y \leq w)*f_Y(y)dy \\
=\int_{-1}^{1}P(X \leq w/y)*f_Y(y)dy \\
=\int_{-1}^{1}F_X(w/y)*f_Y(y)dy \\
[/tex]
For some RV Z, Z~U[-1,1], [itex] F_Z(z)=(z+1)/2 [/itex]
And the density of Y is constant (1/2).
So the integral becomes
[tex]
\int_{-1}^{1} \frac{\frac{w}{y}+1}{2}*\frac{1}{2}dy \\
=\int_{-1}^{1} \frac{w}{4y}+\frac{1}{4} dy \\
=\frac{w}{4}log(y)+\frac{y}{4} \rvert_{-1}^1 \\
[/tex]
which obviously has a problem.
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