Let X be a random variable with the following probability distribution: X f(x) -2 0.5 3 0.3 Find the standard deviation of X. The standard deviation is 5 0.2 4 ... (Round to two decimal places as needed.)
Let X be a random variable with the following probability distribution: X f(x) -2 0.5 3 0.3 Find the standard deviation of X. The standard deviation is 5 0.2 4 ... (Round to two decimal places as needed.)
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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18. Please answer the following question.
![**Probability Distribution and Standard Deviation Calculation**
Let \( X \) be a random variable with the following probability distribution:
\[
\begin{array}{c|ccc}
x & -2 & 3 & 5 \\
\hline
f(x) & 0.5 & 0.3 & 0.2 \\
\end{array}
\]
**Task:** Find the standard deviation of \( X \).
---
**Solution:**
1. **Mean Calculation (\(\mu\)):**
\[
\mu = \sum (x \cdot f(x)) = (-2 \cdot 0.5) + (3 \cdot 0.3) + (5 \cdot 0.2)
\]
2. **Variance Calculation (\(\sigma^2\)):**
\[
\sigma^2 = \sum ((x - \mu)^2 \cdot f(x))
\]
3. **Standard Deviation (\(\sigma\)):**
\[
\sigma = \sqrt{\sigma^2}
\]
Round the final answer to two decimal places as needed.
**Final Answer:**
The standard deviation is \(\Box\). (Round to two decimal places as needed.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fabd5e741-0235-409f-80de-883f5f0e5da7%2F20aaa304-ea1e-4cba-94f9-446ca688877f%2Fu6oh5ar_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Probability Distribution and Standard Deviation Calculation**
Let \( X \) be a random variable with the following probability distribution:
\[
\begin{array}{c|ccc}
x & -2 & 3 & 5 \\
\hline
f(x) & 0.5 & 0.3 & 0.2 \\
\end{array}
\]
**Task:** Find the standard deviation of \( X \).
---
**Solution:**
1. **Mean Calculation (\(\mu\)):**
\[
\mu = \sum (x \cdot f(x)) = (-2 \cdot 0.5) + (3 \cdot 0.3) + (5 \cdot 0.2)
\]
2. **Variance Calculation (\(\sigma^2\)):**
\[
\sigma^2 = \sum ((x - \mu)^2 \cdot f(x))
\]
3. **Standard Deviation (\(\sigma\)):**
\[
\sigma = \sqrt{\sigma^2}
\]
Round the final answer to two decimal places as needed.
**Final Answer:**
The standard deviation is \(\Box\). (Round to two decimal places as needed.)
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