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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 \( X + Y \)?

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#### 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:
\[
X + Y \sim \text{Poi}(\lambda + \kappa).
\]

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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!

Moment Generating Functions. Let X be a random variable, and recall the moment generating function M_X is defined by M_X(t) = E[e^(tX)] for t in R. (a) Compute M_X when (i) X ~ N(μ, σ^2), (ii) X ~ Poi(λ). (b) If X ~ N(μ, σ^2) and Y ~ N(ν, θ^2) are independent normal random variables with respective means μ, ν and variances σ^2, θ^2, and α, β in R, what is the distribution of αX + βY? (c) If X ~ Poi(λ) and Y ~ Poi(κ) are independent Poisson random variables, what is the distribution of X + Y?

Learn how to compute the moment generating functions (MGF) for normal and Poisson distributions, and explore the distributions of sums of independent random variables. This detailed explanation covers the key formulas for MGFs, as well as the resulting distributions for sums of independent normal and Poisson variables.

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