6. X has a discrete uniform distribution fix) = 1/k, x= 1, 2, ., k. i. Show that the moment generations function (MGF) is e(- kC1-et ii. Using the MGF show that E(X) =
6. X has a discrete uniform distribution fix) = 1/k, x= 1, 2, ., k. i. Show that the moment generations function (MGF) is e(- kC1-et ii. Using the MGF show that E(X) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Probability & Statistics 1
![**Problem 6: Discrete Uniform Distribution**
X has a discrete uniform distribution.
Given:
\[ f(x) = \frac{1}{k}, \, x = 1, 2, \ldots, k \]
**Tasks:**
i. **Show that the moment generating function (MGF) is:**
\[
\frac{e^t(1-e^{kt})}{k(1-e^t)}
\]
ii. **Using the MGF, show that \( E(X) = \frac{k+1}{2} \)**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff5203d42-0da6-4b1b-a1a9-5b37798d2cd6%2Fa65ec5c7-7186-4569-a7b1-de9113da55ff%2Fvat5z2j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 6: Discrete Uniform Distribution**
X has a discrete uniform distribution.
Given:
\[ f(x) = \frac{1}{k}, \, x = 1, 2, \ldots, k \]
**Tasks:**
i. **Show that the moment generating function (MGF) is:**
\[
\frac{e^t(1-e^{kt})}{k(1-e^t)}
\]
ii. **Using the MGF, show that \( E(X) = \frac{k+1}{2} \)**
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