For some t > 1, let X be a random variable taking the values 0 and t, with probabilities Then E[X] = 1 and Var(X) = t−1. Consider the probability P(|X − 1| > a). a. Verify the following: if t = 10 and...

For some t > 1, let X be a random variable taking the values 0 and t, with probabilities

Then E[X] = 1 and Var(X) = t−1. Consider the probability P(|X − 1| > a).

a. Verify the following: if t = 10 and a = 8 then P(|X − 1| > a)=1/10 and Chebyshev’s inequality gives an upper bound for this probability of 9/64. The difference is 9/64 − 1/10 ≈ 0.04. We will say that for t = 10 the Chebyshev gap for X at a = 8 is 0.04.

b. Compute the Chebyshev gap for t = 10 at a = 5 and at a = 10.

c. Can you find a gap smaller than 0.01, smaller than 0.001, smaller than 0.0001?

d. Do you think one could improve Chebyshev’s inequality, i.e., find an upper bound closer to the true probabilities?