Give me a 3 PMFS where i have to derrive the MGF or Var(X) using MGF with solutions

Sure! Here are three distinct probability mass functions (PMFs) along with their required tasks (either deriving the Moment Generating Function (MGF) or finding the Variance using MGF):

### **1. Bernoulli Distribution**
PMF:
\[
P(X = 1) = p, \quad P(X = 0) = 1 - p, \quad \text{where } 0 \leq p \leq 1
\]

**Task: Derive the MGF of \(X\) and find \(Var(X)\) using the MGF.**

#### **Solution:**
The MGF of a random variable \(X\) is defined as:
\[
M_X(t) = \mathbb{E}[e^{tX}]
\]

For a Bernoulli distribution:
\[
M_X(t) = (1-p)e^{t \cdot 0} + p e^{t \cdot 1} = 1 - p + p e^t
\]

**First derivative of MGF:**
\[
M_X'(t) = p e^t
\]
Evaluate \( M_X'(t) \) at \(t = 0\) to get the expected value \( \mathbb{E}[X] \):
\[
\mathbb{E}[X] = M_X'(0) = p
\]

**Second derivative of MGF:**
\[
M_X''(t) = p e^t
\]
Evaluate \( M_X''(t) \) at \(t = 0\) to get the second moment \( \mathbb{E}[X^2] \):
\[
\mathbb{E}[X^2] = M_X''(0) = p
\]

The variance is:
\[
Var(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2 = p - p^2 = p(1 - p)
\]

---

### **2. Poisson Distribution**
PMF:
\[
P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \quad k = 0, 1, 2, \dots, \lambda > 0
\]

**Task: Derive the MGF of \(X\) and find \(Var(X)\) using the MGF.**

#### **Solution:**
The MGF of a Poisson random variable \(X\) is:
\[
M_X(t) = \mathbb{E}[e^{tX}] = \

Give me a 3 PMFs where I have to derive the MGF or Var(X) using MGF with solutions.

Explore how to derive the Moment Generating Function (MGF) and calculate variance using MGF for Bernoulli and Poisson distributions. This solution covers key probability concepts and detailed steps for finding MGF and variance using standard formulas and theorems.

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