1. Using Chebyshevs inequality: (a) Show that any random variable X with mean and variance 2 satisfies P(|X | > a) 1

1. Using Chebyshev’s inequality:

(a) Show that any random variable X with mean μ and variance σ2 satisfies P(|X − μ| > aσ) ≤

1 a2 , that is, that the probability X differs from its mean by more than a standard deviations cannot exceed 1/a2.

(b) Supposing X ∼ N(μ, σ2), compute the exact probability that X differs from its mean by more than a standard deviations for a = 1, 2, and 3, and compare the exact probabilities with the upper bounds provided by Chebyshev’s inequality.