Determine the probability density function for the following cumulative distribution function. F(x) = 0 0.25x +0.5 0.5x +0.25 1 x < -2 -2 ≤ x < 1 1
Determine the probability density function for the following cumulative distribution function. F(x) = 0 0.25x +0.5 0.5x +0.25 1 x < -2 -2 ≤ x < 1 1
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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![### Determining the Probability Density Function from a Given Cumulative Distribution Function
Given the following cumulative distribution function \( F(x) \):
\[
F(x) =
\begin{cases}
0 & \text{if } x < -2 \\
0.25x + 0.5 & \text{if } -2 \leq x < 1 \\
0.5x + 0.25 & \text{if } 1 \leq x < 1.5 \\
1 & \text{if } 1.5 \leq x
\end{cases}
\]
We aim to determine the probability density function (PDF) and find the value of the PDF at \( x = 0.1 \).
### Steps to Find the Probability Density Function
1. **Identify the Cumulative Distribution Function (CDF) Intervals**:
- \( x < -2 \): \( F(x) = 0 \)
- \(-2 \leq x < 1 \): \( F(x) = 0.25x + 0.5 \)
- \( 1 \leq x < 1.5 \): \( F(x) = 0.5x + 0.25 \)
- \( x \geq 1.5 \): \( F(x) = 1 \)
2. **Determine the PDF by Differentiating the CDF**:
- \( f(x) = \frac{d}{dx} F(x) \)
The derivations for each interval are as follows:
- For \( x < -2 \):
\[
\frac{d}{dx} F(x) = \frac{d}{dx} 0 = 0
\]
- For \( -2 \leq x < 1 \):
\[
\frac{d}{dx} (0.25x + 0.5) = 0.25
\]
- For \( 1 \leq x < 1.5 \):
\[
\frac{d}{dx} (0.5x + 0.25) = 0.5
\]
- For \( x \geq 1.5 \):
\[
\frac{d}{dx} 1 = 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F20c5837f-8a07-4fb0-b110-a8d2b8a51a74%2Fba1cd7db-119c-46d2-983d-6d81f1002e68%2Fmjnn8rh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Determining the Probability Density Function from a Given Cumulative Distribution Function
Given the following cumulative distribution function \( F(x) \):
\[
F(x) =
\begin{cases}
0 & \text{if } x < -2 \\
0.25x + 0.5 & \text{if } -2 \leq x < 1 \\
0.5x + 0.25 & \text{if } 1 \leq x < 1.5 \\
1 & \text{if } 1.5 \leq x
\end{cases}
\]
We aim to determine the probability density function (PDF) and find the value of the PDF at \( x = 0.1 \).
### Steps to Find the Probability Density Function
1. **Identify the Cumulative Distribution Function (CDF) Intervals**:
- \( x < -2 \): \( F(x) = 0 \)
- \(-2 \leq x < 1 \): \( F(x) = 0.25x + 0.5 \)
- \( 1 \leq x < 1.5 \): \( F(x) = 0.5x + 0.25 \)
- \( x \geq 1.5 \): \( F(x) = 1 \)
2. **Determine the PDF by Differentiating the CDF**:
- \( f(x) = \frac{d}{dx} F(x) \)
The derivations for each interval are as follows:
- For \( x < -2 \):
\[
\frac{d}{dx} F(x) = \frac{d}{dx} 0 = 0
\]
- For \( -2 \leq x < 1 \):
\[
\frac{d}{dx} (0.25x + 0.5) = 0.25
\]
- For \( 1 \leq x < 1.5 \):
\[
\frac{d}{dx} (0.5x + 0.25) = 0.5
\]
- For \( x \geq 1.5 \):
\[
\frac{d}{dx} 1 = 0
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