Let f(x)= 4e-** if 0
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![**Instructions:**
Given a probability density function (pdf):
\[ f(x) = 4e^{-4x} \quad \text{if} \quad 0 < x < \infty, \quad \text{and zero otherwise}. \]
You are required to:
a. **Show that \( f(x) \) is a pdf.**
- To demonstrate this, you need to verify two conditions:
1. \( f(x) \geq 0 \) for all \( x \).
2. The integral of \( f(x) \) over its entire range is 1, i.e., \( \int_{0}^{\infty} f(x) \, dx = 1 \).
b. **Find the Moment Generating Function \( M_X(t) \).**
- Calculate the moment generating function (MGF) \( M_X(t) \) of the given pdf, which is defined as \( M_X(t) = E(e^{tX}) \).
c. **Use \( M_X(t) \) to find \( E(X) \).**
- Utilize the MGF to find the expected value \( E(X) \). Recall that the first derivative of the MGF at \( t = 0 \) gives the mean of the distribution, i.e., \( E(X) = M_X'(0) \).
**Steps to follow:**
1. Integrate \( f(x) \) over \( [0, \infty) \) to ensure it equals 1.
2. Compute \( M_X(t) \) by integrating \( e^{tx} f(x) \) over \( [0, \infty) \).
3. Differentiate \( M_X(t) \) and evaluate it at \( t = 0 \) to find \( E(X) \).
This exercise will help solidify your understanding of probability density functions, moment generating functions, and expected values in probability theory.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa512e5ed-0cb1-424e-ac94-5a6d2cf189ee%2F02162d2c-2804-4d88-8b95-30166876a8d3%2F12pls7s.png&w=3840&q=75)
Transcribed Image Text:**Instructions:**
Given a probability density function (pdf):
\[ f(x) = 4e^{-4x} \quad \text{if} \quad 0 < x < \infty, \quad \text{and zero otherwise}. \]
You are required to:
a. **Show that \( f(x) \) is a pdf.**
- To demonstrate this, you need to verify two conditions:
1. \( f(x) \geq 0 \) for all \( x \).
2. The integral of \( f(x) \) over its entire range is 1, i.e., \( \int_{0}^{\infty} f(x) \, dx = 1 \).
b. **Find the Moment Generating Function \( M_X(t) \).**
- Calculate the moment generating function (MGF) \( M_X(t) \) of the given pdf, which is defined as \( M_X(t) = E(e^{tX}) \).
c. **Use \( M_X(t) \) to find \( E(X) \).**
- Utilize the MGF to find the expected value \( E(X) \). Recall that the first derivative of the MGF at \( t = 0 \) gives the mean of the distribution, i.e., \( E(X) = M_X'(0) \).
**Steps to follow:**
1. Integrate \( f(x) \) over \( [0, \infty) \) to ensure it equals 1.
2. Compute \( M_X(t) \) by integrating \( e^{tx} f(x) \) over \( [0, \infty) \).
3. Differentiate \( M_X(t) \) and evaluate it at \( t = 0 \) to find \( E(X) \).
This exercise will help solidify your understanding of probability density functions, moment generating functions, and expected values in probability theory.
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