Gamma distribution question with n independent variables

**lllll** · October 1st 2008, 10:44 PM

I need a some help with the following statement:

Let $Y_1,Y_2,...,Y_n$ be independent r.v., where $Y_i$ has a gamma distribution, with parameters $\alpha_i$ and $\beta$ (where $\beta$ is fixed and the values of $\alpha$ are different). Prove that $U=Y_1+Y_2+...+Y_n$ has a gamma distribution with parameters $\alpha_1+\alpha_2+...+\alpha_n$ and $\beta$

so far I have:

$U=Y_1+Y_2+...+Y_n$

$m_U(t) = m_{Y_1}(t)\times m_{Y_2}(t)\times ... \times m_{Y_n}(t)$

$m_U(t) = (1-\beta)^{-\alpha_1} \times (1-\beta)^{-\alpha_2} \times ... \times (1-\beta)^{-\alpha_n}$

$m_U(t)=\prod^{n}_{i=1} (1-\beta)^{-\alpha_i}= \$

at which point I don't know how to continue it.

**Laurent** · October 1st 2008, 11:40 PM

Why not write: $<br /> m_U(t)=\prod^{n}_{i=1} (1-\beta)^{-\alpha_i}= (1-\beta)^{-\sum_{i=1}^n\alpha_i}<br />$ ?

(By the way, it is not straightforward to show that the moment generating function of a Gamma distribution is $(1-\beta)^{-\alpha}$ , and the most elementary proof of your statement is by change of variable in multiple integrals. But if you were given the moment generating function, you're definitely right using it, since this is way quicker.)

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