Q: Let X be a random variable with pdf given below. Find the constant c such that this is a valid pdf. for 0
Q: Let X be a random variable with pdf given below. Find the constant c such that this is a valid pdf. for 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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Section: Probabilty Density Functions
![**Problem Statement:**
Let \( X \) be a random variable with probability density function (pdf) given below. Find the constant \( c \) such that this is a valid probability density function.
\[
f(x) =
\begin{cases}
cx^2 & \text{for } 0 \leq x < 1, \\
c(2-x)^2 & \text{for } 1 \leq x \leq 2, \\
0 & \text{otherwise.}
\end{cases}
\]
**Explanation:**
The function \( f(x) \) is a piecewise function defined on the interval from 0 to 2. It consists of two parts:
1. For \( 0 \leq x < 1 \), the function is \( cx^2 \).
2. For \( 1 \leq x \leq 2 \), the function is \( c(2-x)^2 \).
Outside the interval [0, 2], the function is zero.
To ensure this is a valid pdf, the integral of \( f(x) \) over its entire range must equal 1.
\[ \int_{-\infty}^{\infty} f(x) \, dx = 1 \]
This requirement translates to:
\[ \int_{0}^{1} cx^2 \, dx + \int_{1}^{2} c(2-x)^2 \, dx = 1 \]
By solving this equation, we can determine the value of \( c \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7b45ec80-7eee-4c10-9c37-d097772a8db5%2Ff9985c9e-fef6-46b9-869e-93846ef93554%2Fwop7w6v_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let \( X \) be a random variable with probability density function (pdf) given below. Find the constant \( c \) such that this is a valid probability density function.
\[
f(x) =
\begin{cases}
cx^2 & \text{for } 0 \leq x < 1, \\
c(2-x)^2 & \text{for } 1 \leq x \leq 2, \\
0 & \text{otherwise.}
\end{cases}
\]
**Explanation:**
The function \( f(x) \) is a piecewise function defined on the interval from 0 to 2. It consists of two parts:
1. For \( 0 \leq x < 1 \), the function is \( cx^2 \).
2. For \( 1 \leq x \leq 2 \), the function is \( c(2-x)^2 \).
Outside the interval [0, 2], the function is zero.
To ensure this is a valid pdf, the integral of \( f(x) \) over its entire range must equal 1.
\[ \int_{-\infty}^{\infty} f(x) \, dx = 1 \]
This requirement translates to:
\[ \int_{0}^{1} cx^2 \, dx + \int_{1}^{2} c(2-x)^2 \, dx = 1 \]
By solving this equation, we can determine the value of \( c \).
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