This image focuses on Moment Generating Functions (MGF) and requests the following:

1. Definition: The moment generating function $M_X(t)$ for a random variable $X$ is defined as: $M_X(t) = \mathbb{E}\left[e^{tX}\right], \quad t \in \mathbb{R}.$ The MGF helps characterize the distribution of the random variable and is particularly useful for calculating moments.

2. Existence: The MGF exists if there is a $\delta > 0$ such that $M_X(t)$ is finite for all $t$ in the interval $|t| < \delta$. Two random variables with the same MGF (if it exists) have the same distribution.

The problems are divided into three parts:

(a) Compute $M_X(t)$ for:

• (i) $X \sim N(\mu, \sigma^2)$: The normal distribution with mean $\mu$ and variance $\sigma^2$.
• (ii) $X \sim \text{Poi}(\lambda)$: The Poisson distribution with rate $\lambda$.

(b) If $X \sim N(\mu, \sigma^2)$ and $Y \sim N(\nu, \theta^2)$ are independent, find the distribution of $\alpha X + \beta Y$ where $\alpha, \beta \in \mathbb{R}$.

(c) If $X \sim \text{Poi}(\lambda)$ and $Y \sim \text{Poi}(\kappa)$ are independent, what is the distribution of $X + Y$?

Solutions:

(a)

1. (i) $X \sim N(\mu, \sigma^2)$: The MGF of a normally distributed random variable is: $M_X(t) = \exp\left(\mu t + \frac{1}{2} \sigma^2 t^2\right).$

2. (ii) $X \sim \text{Poi}(\lambda)$: The MGF of a Poisson-distributed random variable is: $M_X(t) = \exp\left(\lambda (e^t - 1)\right).$

(b)

For independent normal random variables $X \sim N(\mu, \sigma^2)$ and $Y \sim N(\nu, \theta^2)$, the linear combination $\alpha X + \beta Y$ is also normally distributed with: $\alpha X + \beta Y \sim N\left(\alpha \mu + \beta \nu, \alpha^2 \sigma^2 + \beta^2 \theta^2\right).$

(c)

If $X \sim \text{Poi}(\lambda)$ and $Y \sim \text{Poi}(\kappa)$ are independent, the sum $X + Y$ is also Poisson-distributed with: $X + Y \sim \text{Poi}(\lambda + \kappa).$

Would you like further details on any part of the explanation?

Here are some related questions:

1. What is the relationship between the MGF and the cumulant generating function (CGF)?
2. How can the MGF be used to find the moments of a distribution?
3. What happens if two different distributions have the same MGF?
4. How is the MGF related to the characteristic function of a random variable?
5. What are the applications of MGFs in statistical inference?

Tip: Moment generating functions can be powerful tools for deriving distributions of sums of independent random variables!