Chebyshev's inequality .(Section 2.4) states that for any

Chapter 4, Problem 26E

(choose chapter or problem)

Get Unlimited Answers
QUESTION:

Chebyshev’s inequality (Section 2.4) states that for any random variable X with mean μ and variance σ2, and for any positive number k, \(P(|X=\mu| \geq k \sigma) \leq 1 / k^{2}\). Let X ∼ N\(\left(\mu, \sigma^{2}\right)\). Compute \(P(|X-\mu| \geq k \sigma)\) for the values k = 1, 2, and 3. Are the actual probabilities close to the Chebyshev bound of \(1 / k^{2}\), or are they much smaller?

Equation transcription:

Text transcription:

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

(mu, sigma^{2})

P(|X-mu| geq k sigma)

1 / k^{2}

Questions & Answers

QUESTION:

Chebyshev’s inequality (Section 2.4) states that for any random variable X with mean μ and variance σ2, and for any positive number k, \(P(|X=\mu| \geq k \sigma) \leq 1 / k^{2}\). Let X ∼ N\(\left(\mu, \sigma^{2}\right)\). Compute \(P(|X-\mu| \geq k \sigma)\) for the values k = 1, 2, and 3. Are the actual probabilities close to the Chebyshev bound of \(1 / k^{2}\), or are they much smaller?

Equation transcription:

Text transcription:

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

(mu, sigma^{2})

P(|X-mu| geq k sigma)

1 / k^{2}

ANSWER:

Solution 26E

Step1 of 2:

We have chebyshev’s inequality and it states that for any random variable X with mean and variance and for any number of k,

Also we have a random variable X it follows normal distribution with mean “n” and

variance “” .

We need to compute for the values for k = 1, 2 and 3 also we have to check whether the actual probabilities are close to chebyshev’s bound or are they much smaller.


 Step2 of 2:

From the given informa we have chebyshev’s inequality

There are 68% of the population is in the interval therefore

We need to compute for k = 1, 2, 3.

Add to cart


Study Tools You Might Need

Not The Solution You Need? Search for Your Answer Here:

×

Login

Login or Sign up for access to all of our study tools and educational content!

Forgot password?
Register Now

×

Register

Sign up for access to all content on our site!

Or login if you already have an account

×

Reset password

If you have an active account we’ll send you an e-mail for password recovery

Or login if you have your password back