Question

You’re waiting for Caltrain. Suppose that the waiting times have a mean of 12 minutes and a standard deviation of 3 minutes. Use the Chebyshev inequality to answer each of the following questions:

a) What is the largest possible probability that you’ll end up waiting either less than 6 minutes or more than 18 minutes for the train?

b) What is the smallest possible probability that you’ll wait between 6 and 18 minutes for the train?

c) What is the smallest possible probability that you’ll wait between 3 and 21 minutes for the train?

d) What is the smallest possible probability that you’ll wait between 0 and 24 minutes for the train?

e) Based on your answer to part (d), what is the largest possible probability that you’ll need to wait more than 24 minutes for the train? Why is this answer so dramatically different from your answer to #1c above?

Answer 1

Chebsyhev's inequality states that no more than a certain fraction of values are more than a certain distance from the mean.

rac{1}{k^2} percent of values can be more than k times teh standard deviations away from the mean.

The statement of the Chebsyhev's inequality

P(mu-ksigma <X<mu-ksigma)le 1- rac{1}{k^2}

X is a random variable with mean mu and standard deviation sigma

We are given
mu = 12
sigma = 3

Question 1
Let k = 2

Substituing the values

Hence probability that you’ll end up waiting either less than 6 minutes or more than 18 minutes for the train is 0.25

Question 2
Let k = 2

Question 3

Let k = 3

Smallest possible probability that you’ll wait between 3 and 21 minutes for the train 0.89

Question 4

Let k = 4

Smallest possible probability that you’ll wait between 0 and 24 minutes for the train =0.9375