Find the constant C such that p(x) is a probability function. Compute the probability P(1.5≤X ≤2).
Find the constant C such that p(x) is a probability function. Compute the probability P(1.5≤X ≤2).
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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Question
![Let \( p(x) = C \cdot \frac{1}{x} \) on \([1 \leq X \leq 2]\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f3d9715-5083-4150-a4a6-02c5c7b14656%2F6d0e60d6-c7f9-49a9-a69d-c3517b51b127%2Fghiuqn_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( p(x) = C \cdot \frac{1}{x} \) on \([1 \leq X \leq 2]\).
![### Problem Statement
**Objective:**
1. Find the constant \( C \) such that \( p(x) \) is a probability function.
2. Compute the probability \( P(1.5 \leq X \leq 2) \).
### Solution Steps
To solve the problem, follow these steps:
#### Step 1: Determine the constant \( C \)
To find the constant \( C \) that makes \( p(x) \) a probability function, we need to ensure that the total area under the probability density function (pdf) \( p(x) \) over the possible values of \( x \) is equal to 1.
#### Step 2: Calculate the probability \( P(1.5 \leq X \leq 2) \)
Once we have the value of \( C \), we can integrate the pdf \( p(x) \) from 1.5 to 2 to find this probability.
### Detailed Explanation
1. **Finding the constant \( C \):**
- This requires integrating \( p(x) \) over the domain \( x \) where \( p(x) \) is defined, and setting the integral equal to 1.
\[
\int_{a}^{b} p(x) \, dx = 1
\]
- Solve for \( C \).
2. **Computing the probability \( P(1.5 \leq X \leq 2) \):**
- Integrate the probability density function \( p(x) \) from 1.5 to 2.
\[
P(1.5 \leq X \leq 2) = \int_{1.5}^{2} p(x) \, dx
\]
- Use the value of \( C \) obtained from the previous step to evaluate this integral.
### Graphical Explanation (if any Graph or Diagram)
- **Graph Consideration:**
- If there are graphs depicting the pdf \( p(x) \):
- The x-axis represents the variable \( x \).
- The y-axis represents the probability density \( p(x) \).
- The area under the curve of \( p(x) \) between \( x = 1.5 \) and \( x = 2 \) represents the probability \( P(1.5 \leq X \leq 2) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f3d9715-5083-4150-a4a6-02c5c7b14656%2F6d0e60d6-c7f9-49a9-a69d-c3517b51b127%2Fxbzcvnu_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Objective:**
1. Find the constant \( C \) such that \( p(x) \) is a probability function.
2. Compute the probability \( P(1.5 \leq X \leq 2) \).
### Solution Steps
To solve the problem, follow these steps:
#### Step 1: Determine the constant \( C \)
To find the constant \( C \) that makes \( p(x) \) a probability function, we need to ensure that the total area under the probability density function (pdf) \( p(x) \) over the possible values of \( x \) is equal to 1.
#### Step 2: Calculate the probability \( P(1.5 \leq X \leq 2) \)
Once we have the value of \( C \), we can integrate the pdf \( p(x) \) from 1.5 to 2 to find this probability.
### Detailed Explanation
1. **Finding the constant \( C \):**
- This requires integrating \( p(x) \) over the domain \( x \) where \( p(x) \) is defined, and setting the integral equal to 1.
\[
\int_{a}^{b} p(x) \, dx = 1
\]
- Solve for \( C \).
2. **Computing the probability \( P(1.5 \leq X \leq 2) \):**
- Integrate the probability density function \( p(x) \) from 1.5 to 2.
\[
P(1.5 \leq X \leq 2) = \int_{1.5}^{2} p(x) \, dx
\]
- Use the value of \( C \) obtained from the previous step to evaluate this integral.
### Graphical Explanation (if any Graph or Diagram)
- **Graph Consideration:**
- If there are graphs depicting the pdf \( p(x) \):
- The x-axis represents the variable \( x \).
- The y-axis represents the probability density \( p(x) \).
- The area under the curve of \( p(x) \) between \( x = 1.5 \) and \( x = 2 \) represents the probability \( P(1.5 \leq X \leq 2) \).
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