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 $Y$?
Tell me more
×
Mathematics Stack Exchange is a question and answer site for
people studying math at any level and professionals in related fields. It's 100% free, no registration required.
|
|
||||
|
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 |
||||
|
homeworktag. 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