d a d Hint Apply the usual Markov inequality to the new non negative random

# D a d hint apply the usual markov inequality to the

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d a - d Hint: Apply the usual Markov inequality to the new non-negative random variable a - X . 15. (Optional) [Two Independent Random Variables] Let X and Y be independent random variables, uniformly distributed on [0 , 2]. (a) Find the mean and variance of XY . (b) Calculate the probability Pr( XY 1). You may use, without proof, the fact that for a uniform random variable on [ a, b ], the variance is ( b - a ) 2 / 12. 16. (Optional) [Convergence in Mean Square] Suppose X n X in mean square as n → ∞ and E [ X 4 n ] < for all n . Does X 2 n necessarily converge in mean square as n → ∞ ? 17. (Optional) [Laws of Large Numbers] (a) State Chebychev’s Inequality. (b) Suppose { X i } i =1 is a sequence of uncorrelated random variables (i.e., E [ X i X j ] = E [ X i ] E [ X j ] for i 6 = j ), each of which has finite mean, and assume that for all n , Var( X n ) M < . Define μ n = 1 n n X i =1 E [ X i ] . and suppose that μ n μ as n → ∞ and | μ | < . Let S n = n i =1 X i . Show that S n /n μ in probability. 2

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• Summer '19
• Probability theory, Gallager

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