# Gamma distribution - find the mgf

Let $X_1, X_2$ be two independent random variables having gamma distributions with parameters $\alpha_1 = 3$, $\beta_1 = 3$ and $\alpha_2 = 5$, $\beta_2 = 1$, respectively.

(a) Find the mgf of $Y = 2X_1 + 6X_2$.

(b) What is the distribution of $Y$?

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 If this is homework, please add the homework tag. Hint: Depending on which definition of Gamma random variables you are using, $2X_1$ and $6X_2$ might be independent Gamma random variables with parameter $\beta=6$. What do you know about the sum of independent Gamma random variables with the same scale (or rate) parameter? – Dilip Sarwate Nov 13 '12 at 4:15

gamma dist has two parameters: shape $\alpha$ and scale $\beta$, where you can add shape and multiple scale.

for your probblem, $Y=2X1+6X2 \leadsto \gamma(3,6)+\gamma(5,6)=\gamma(8,6)$, thus the mgf is $(1-6t)^{-8}$

it may be helpful to see the reference below

http://en.wikipedia.org/wiki/Gamma_distribution

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 how are you getting that both $\beta_1$ and $\beta_2$ are 6? It says that they are 3 and 1 respective. And $\beta=\frac{1}{\theta}$ so shouldn't it be $\gamma(3,\frac{1}{6})$? – Christopher Ernst Apr 23 at 2:16