Determine the probability density function for the following cumulative distribution function. 0 F(x) = x < -2 0.25x +0.5-2 < x < 1 0.5x +0.25 1 < x < 1.5 1.5 ≤ x Find the value of the probability density function at x = -1.5.
Determine the probability density function for the following cumulative distribution function. 0 F(x) = x < -2 0.25x +0.5-2 < x < 1 0.5x +0.25 1 < x < 1.5 1.5 ≤ x Find the value of the probability density function at x = -1.5.
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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![**Title: Understanding Probability Density Functions from Cumulative Distribution Functions**
**Introduction to Cumulative Distribution Function (CDF) and Probability Density Function (PDF)**
In this lesson, we will determine the probability density function (PDF) from a given cumulative distribution function (CDF). Understanding the relationship between these two functions is crucial in probability and statistics as they provide a comprehensive picture of the distribution of a random variable.
**Given Cumulative Distribution Function (CDF) F(x):**
\[
F(x) =
\begin{cases}
0 & x < -2 \\
0.25x + 0.5 & -2 \leq x < 1 \\
0.5x + 0.25 & 1 \leq x < 1.5 \\
1 & 1.5 \leq x
\end{cases}
\]
**Objective:**
1. Determine the probability density function (PDF) for the given CDF.
2. Find the value of the probability density function at \( x = -1.5 \).
**Detailed Steps to Find the PDF:**
To derive the probability density function \( f(x) \) from the cumulative distribution function \( F(x) \), we need to compute the derivative of \( F(x) \). The PDF is the rate of change of CDF with respect to \( x \).
**1. Identify Different Regions:**
\[
F(x) =
\begin{cases}
0 & \text{for } x < -2 \\
0.25x + 0.5 & \text{for } -2 \leq x < 1 \\
0.5x + 0.25 & \text{for } 1 \leq x < 1.5 \\
1 & \text{for } 1.5 \leq x
\end{cases}
\]
**2. Compute the derivative for each region:**
- For \( x < -2 \):
\[
F(x) = 0 \Rightarrow f(x) = \frac{d}{dx}[0] = 0
\]
- For \( -2 \leq x < 1 \):
\[
F(x) = 0.25x + 0.5 \Rightarrow f(x) = \frac{d}{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F20c5837f-8a07-4fb0-b110-a8d2b8a51a74%2F5a79197a-2d6e-470b-9f89-0e882f15f79a%2F6mmv44n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Probability Density Functions from Cumulative Distribution Functions**
**Introduction to Cumulative Distribution Function (CDF) and Probability Density Function (PDF)**
In this lesson, we will determine the probability density function (PDF) from a given cumulative distribution function (CDF). Understanding the relationship between these two functions is crucial in probability and statistics as they provide a comprehensive picture of the distribution of a random variable.
**Given Cumulative Distribution Function (CDF) F(x):**
\[
F(x) =
\begin{cases}
0 & x < -2 \\
0.25x + 0.5 & -2 \leq x < 1 \\
0.5x + 0.25 & 1 \leq x < 1.5 \\
1 & 1.5 \leq x
\end{cases}
\]
**Objective:**
1. Determine the probability density function (PDF) for the given CDF.
2. Find the value of the probability density function at \( x = -1.5 \).
**Detailed Steps to Find the PDF:**
To derive the probability density function \( f(x) \) from the cumulative distribution function \( F(x) \), we need to compute the derivative of \( F(x) \). The PDF is the rate of change of CDF with respect to \( x \).
**1. Identify Different Regions:**
\[
F(x) =
\begin{cases}
0 & \text{for } x < -2 \\
0.25x + 0.5 & \text{for } -2 \leq x < 1 \\
0.5x + 0.25 & \text{for } 1 \leq x < 1.5 \\
1 & \text{for } 1.5 \leq x
\end{cases}
\]
**2. Compute the derivative for each region:**
- For \( x < -2 \):
\[
F(x) = 0 \Rightarrow f(x) = \frac{d}{dx}[0] = 0
\]
- For \( -2 \leq x < 1 \):
\[
F(x) = 0.25x + 0.5 \Rightarrow f(x) = \frac{d}{
![### Finding the Probability Density Function
#### Problem Statement:
Determine the probability density function for the following cumulative distribution function (CDF):
\[
F(x) =
\begin{cases}
0 & \text{if } x < -2 \\
0.25x + 0.5 & \text{if } -2 \le x < 1 \\
0.5x + 0.25 & \text{if } 1 \le x < 1.5 \\
1 & \text{if } 1.5 \le x
\end{cases}
\]
#### Task:
Find the value of the probability density function at \( x = 0.1 \).
---
In this problem, you are given a piecewise cumulative distribution function (CDF) \( F(x) \) and need to find its corresponding probability density function (PDF).
To find the PDF from a given CDF, remember that the PDF is the derivative of the CDF. This process involves differentiating the piecewise functions defined by \( F(x) \) over their respective intervals.
#### Step-by-Step Solution:
1. **Identify the intervals in the CDF:**
- \( x < -2 \)
- \( -2 \le x < 1 \)
- \( 1 \le x < 1.5 \)
- \( 1.5 \le x \)
2. **Differentiate the CDF for each interval:**
- For \( x < -2 \):
\[
F(x) = 0 \implies f(x) = \frac{d}{dx}(0) = 0
\]
- For \( -2 \le x < 1 \):
\[
F(x) = 0.25x + 0.5 \implies f(x) = \frac{d}{dx}(0.25x + 0.5) = 0.25
\]
- For \( 1 \le x < 1.5 \):
\[
F(x) = 0.5x + 0.25 \implies f(x) = \frac{d}{dx}(0.5x + 0.25) = 0.5
\]
- For \( 1.5 \le x \):
\[
F(x)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F20c5837f-8a07-4fb0-b110-a8d2b8a51a74%2F5a79197a-2d6e-470b-9f89-0e882f15f79a%2Fytnqbp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the Probability Density Function
#### Problem Statement:
Determine the probability density function for the following cumulative distribution function (CDF):
\[
F(x) =
\begin{cases}
0 & \text{if } x < -2 \\
0.25x + 0.5 & \text{if } -2 \le x < 1 \\
0.5x + 0.25 & \text{if } 1 \le x < 1.5 \\
1 & \text{if } 1.5 \le x
\end{cases}
\]
#### Task:
Find the value of the probability density function at \( x = 0.1 \).
---
In this problem, you are given a piecewise cumulative distribution function (CDF) \( F(x) \) and need to find its corresponding probability density function (PDF).
To find the PDF from a given CDF, remember that the PDF is the derivative of the CDF. This process involves differentiating the piecewise functions defined by \( F(x) \) over their respective intervals.
#### Step-by-Step Solution:
1. **Identify the intervals in the CDF:**
- \( x < -2 \)
- \( -2 \le x < 1 \)
- \( 1 \le x < 1.5 \)
- \( 1.5 \le x \)
2. **Differentiate the CDF for each interval:**
- For \( x < -2 \):
\[
F(x) = 0 \implies f(x) = \frac{d}{dx}(0) = 0
\]
- For \( -2 \le x < 1 \):
\[
F(x) = 0.25x + 0.5 \implies f(x) = \frac{d}{dx}(0.25x + 0.5) = 0.25
\]
- For \( 1 \le x < 1.5 \):
\[
F(x) = 0.5x + 0.25 \implies f(x) = \frac{d}{dx}(0.5x + 0.25) = 0.5
\]
- For \( 1.5 \le x \):
\[
F(x)
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