Let X ~ NB(r, p). Find M,(1). а. Use M,(t) to find E(X). Use M, (t) to find V(X). с. b.

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The given problem involves working with a negative binomial distribution. Here is a transcription suitable for an educational website:

---

### Negative Binomial Distribution Problem

**Problem Statement:**

Let \( X \sim \text{NB}(1, p) \).

**Tasks:**
a. Find \( M_X(t) \).
b. Use \( M_X(t) \) to find \( E(X) \).
c. Use \( M_X(t) \) to find \( V(X) \).

**Details:**

1. **Finding the Moment-Generating Function \( M_X(t) \):**
   \( M_X(t) \) represents the moment-generating function of the random variable \( X \). For a negative binomial distribution with parameters \( r = 1 \) and probability \( p \), determine the expression for \( M_X(t) \).

2. **Using \( M_X(t) \) to Find the Expected Value \( E(X) \):**
   By using the moment-generating function \( M_X(t) \), derive the expected value \( E(X) \) of the random variable \( X \).

3. **Using \( M_X(t) \) to Find the Variance \( V(X) \):**
   Similarly, use the moment-generating function \( M_X(t) \) to compute the variance \( V(X) \) of the random variable \( X \).

---

This problem requires understanding of the negative binomial distribution, specifically when the distribution is parameterized as \( NB(1, p) \), and applying properties of moment-generating functions to derive key statistical properties such as expectation and variance.
Transcribed Image Text:The given problem involves working with a negative binomial distribution. Here is a transcription suitable for an educational website: --- ### Negative Binomial Distribution Problem **Problem Statement:** Let \( X \sim \text{NB}(1, p) \). **Tasks:** a. Find \( M_X(t) \). b. Use \( M_X(t) \) to find \( E(X) \). c. Use \( M_X(t) \) to find \( V(X) \). **Details:** 1. **Finding the Moment-Generating Function \( M_X(t) \):** \( M_X(t) \) represents the moment-generating function of the random variable \( X \). For a negative binomial distribution with parameters \( r = 1 \) and probability \( p \), determine the expression for \( M_X(t) \). 2. **Using \( M_X(t) \) to Find the Expected Value \( E(X) \):** By using the moment-generating function \( M_X(t) \), derive the expected value \( E(X) \) of the random variable \( X \). 3. **Using \( M_X(t) \) to Find the Variance \( V(X) \):** Similarly, use the moment-generating function \( M_X(t) \) to compute the variance \( V(X) \) of the random variable \( X \). --- This problem requires understanding of the negative binomial distribution, specifically when the distribution is parameterized as \( NB(1, p) \), and applying properties of moment-generating functions to derive key statistical properties such as expectation and variance.
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