Suppose I have two Gamma rvs $X1\sim G(m1,\Omega1)$ and $X2\sim G(m2,\Omega2)$ and both are i.i.d. Then how to derive the PDF of product of $X1\cdot X2$. Any help in this regard is highly appreciated.
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1How are they i.i.d. with different parameters? Are $m_1 = m_2$ and $\Omega_1 = \Omega_2$? Do you just mean they are independent? – Gregory Mar 12 '21 at 17:52
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Yes $m1=m2$ and $\Omega1=\Omega2$ – Pranu Mar 12 '21 at 17:56
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This is a special case of PDF of the product of two independent Gamma random variables. I think you should be able to derive the result using $$f_{X_1\cdot X_2}(y)=\int_{-\infty}^{\infty} f_{X_1}(x)f_{X_2}(y/x)\cdot \frac{1}{|x|}~dx.$$ – projectilemotion Mar 12 '21 at 17:58