Chebyshevs inequality (Section 2.4) states that for any random variable X with mean and

Chapter 4, Problem 26

(choose chapter or problem)

Chebyshev's inequality (Section  states that for any random variable  with mean \(\mu\) and variance \(\sigma^{2}\), and for any positive number \(k, P(|X-\mu| \geq k \sigma) \leq 1 / k^{2}\). Let \(X \sim N\left(\mu, \sigma^{2}\right)\). Compute \(P(|X-\mu| \geq k \sigma)\) for the values \(k=1,2, \text { and } 3\). Are the actual probabilities close to the Chebyshev bound of

\(1 / k^{2}\), or are they much smaller?

Equation Transcription:

Text Transcription:

\mu

\sigma^{2}

k, P(|X-\mu| \geq k \sigma) \leq 1 / k^{2}

X \sim N\(\mu, \sigma^{2}\)

P(|X-\mu| \geq k \sigma)

k=1,2,  and  3

1 / k^{2}