The number of customers visiting a store during a day is a random variable with mean EX = 100 and variance Var(X) = 225.

1. Using Chebyshev's inequality, find an upper bound for having more than 120 or less than 80 customers in a day. That is, find an upper bound on

2. Using the one-sided Chebyshev inequality (Problem 21), find an upper bound for having more than 120 customers in a day.

Problem 21

Let X be a random variable with EX = 0 and Var(X) = σ². We would like to prove that for any a > 0, we have

This inequality is sometimes called the one-sided Chebyshev inequality. One way to show this is to use P(X ≥ a) = P(X +c ≥ a+c) for any constant c ∈ R.

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P(X ≤ 80 or X ≥ 120).