Proof: Let z be a continuous random variable o² = E[(x-μ)²] = (x-μ)² f(x)dr 08 Cl-ka 8 - ko FL " + ko rutko (2-0)²1 (2) de + f (x-µ)²1(2)dz + (x-μ)² (2-14) -ka μ+ko
Proof: Let z be a continuous random variable o² = E[(x-μ)²] = (x-μ)² f(x)dr 08 Cl-ka 8 - ko FL " + ko rutko (2-0)²1 (2) de + f (x-µ)²1(2)dz + (x-μ)² (2-14) -ka μ+ko
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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I’m having issues understanding the first 2 lines of the proof, variance, E and how it gave the integration . Can you pls write a preliminary explaining those? Thanks.
![### Chebyshev’s Inequality Proof
**Statement:**
\[ i.e \quad P(|x - \mu| \geq k \sigma) \leq \frac{1}{k^2} \]
**Proof:**
Let \( x \) be a continuous random variable.
\[
\sigma^2 = E[(x - \mu)^2]
\]
\[
= \int_{-\infty}^{\infty} (x - \mu)^2 f(x) \, dx
\]
The range is divided into three parts: from \(-\infty\) to \(\mu - k\sigma\), from \(\mu - k\sigma\) to \(\mu + k\sigma\), and from \(\mu + k\sigma\) to \(+\infty\).
\[
= \int_{-\infty}^{\mu - k\sigma} (x - \mu)^2 f(x) \, dx + \int_{\mu-k\sigma}^{\mu+k\sigma} (x-\mu)^2 f(x) \, dx + \int_{\mu+k\sigma}^{\infty} (x-\mu)^2 f(x) \, dx
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F79599c56-a340-49a0-b0ff-829b3947a798%2Fde61d447-9e7d-4eb2-b87e-7c7e48146bdd%2Fb3ae1c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Chebyshev’s Inequality Proof
**Statement:**
\[ i.e \quad P(|x - \mu| \geq k \sigma) \leq \frac{1}{k^2} \]
**Proof:**
Let \( x \) be a continuous random variable.
\[
\sigma^2 = E[(x - \mu)^2]
\]
\[
= \int_{-\infty}^{\infty} (x - \mu)^2 f(x) \, dx
\]
The range is divided into three parts: from \(-\infty\) to \(\mu - k\sigma\), from \(\mu - k\sigma\) to \(\mu + k\sigma\), and from \(\mu + k\sigma\) to \(+\infty\).
\[
= \int_{-\infty}^{\mu - k\sigma} (x - \mu)^2 f(x) \, dx + \int_{\mu-k\sigma}^{\mu+k\sigma} (x-\mu)^2 f(x) \, dx + \int_{\mu+k\sigma}^{\infty} (x-\mu)^2 f(x) \, dx
\]
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