Q2Multiple correct3 Marks10 May 2026

Let $X$ be a discrete random variable with PMF as follows:\nSuppose $X_1, X_2, X_3 \sim \text{i.i.d. } X$. Define a new random variable $Y = X_1 + X_2 + X_3$. Which of the following option(s) is (are) true about the moment generating function of the random variable $Y$?

A

$M_Y(\lambda) = \frac{e^{-\lambda} + 2 + 2e^{2\lambda}}{5}$

B

$M_Y(\lambda) = \left(\frac{e^{-\lambda} + 2 + 2e^{2\lambda}}{5}\right)^3$

C

$M_Y(\lambda) = M_{X_1}(\lambda) + M_{X_2}(\lambda) + M_{X_3}(\lambda)$

D

$M_Y(\lambda) = M_{X_1}(\lambda) \times M_{X_2}(\lambda) \times M_{X_3}(\lambda)$