1 Inequality Practice
(a) X is a random variable such that X −5 and E[X] = −3. Find an upper bound for the probability of X being greater than or equal to −1.
(b) You roll a die 100 times. Let Y be the sum of the numbers that appear on the die throughout the 100 rolls. Use Chebyshev’s inequality to bound the probability of the sum Y being greater than 400 or less than 300.
2 Tightness of Inequalities
(a) Show by example that Markov’s inequality is tight; that is, show that given k 0, there exists a discrete non-negative random variable X such that P(X ≥ k) = E[X]/k.
(b) Show by example that Chebyshev’s inequality is tight; that is, show that given k ≥ 1, there exists a random variable X such that P(|X −E[X]| ≥ kσ) = 1/k2, where σ2 = varX.
(c) Show that there is no non-negative discrete random variable X 6= 0, that takes values in some finite set {ν1,...,νN}, such that for all k 0, Markov’s inequality is tight; that is, P(X ≥ k) = E[X]/k.
3 Working with the Law of Large Numbers
(a) A fair coin is tossed and you win a prize if there are more than 60% heads. Which is better: 10 tosses or 100 tosses? Explain.
(b) A fair coin is tossed and you win a prize if there are more than 40% heads. Which is better: 10 tosses or 100 tosses? Explain.
(c) A coin is tossed and you win a prize if there are between 40% and 60% heads. Which is better: 10 tosses or 100 tosses? Explain.
(d) A coin is tossed and you win a prize if there are exactly 50% heads. Which is better: 10 tosses or 100 tosses? Explain.
