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2026-04-01 06:06:04.1775023564
Compute the PDF, CDF and expected value of variables
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Density function for 2. (using convolution) is $f_V(v)=\alpha\beta\int\limits_0^ve^{-\beta x}e^{-\alpha(v-x)}dx=\frac{\alpha\beta}{\alpha-\beta}(e^{-\beta v}-e^{-\alpha v})$.
Density function for 3. $F_W(w)=P(W\le w)=P(X^2\le w)=P(X\le\sqrt{w})$$=1-e^{-\alpha \sqrt{w}}$, giving $f_W(w)=\frac{\alpha}{2\sqrt{w}}e^{-\alpha \sqrt{w}}$.
I presume you can compute expectations.