4. Comparing two bounds. (a) (2 pts) State Markov's inequality. (b) (2 pts) Let Y be a random variable with moment-generating function My. Use Markov's inequality to show that, for any t > 0, my 2 a) g e'mMy(t). Hint: ]P(Y 2 a) = 1P(e\"' 2 e\") for any t > 0. (c) (2 pts) As the inequality in (b) holds for all t > 0, we are free to minimize e'mMy(t) over t > 0 to get the smallest upper bound on P(Y 2 a). Show that the upper bound in (b) is smallest when t satises d EMY\") = GMy(t). Hint: The minimum of e'mMy(t) occurs when its derivative equals zero. (d) (4 pts) Let Y be the number of heads in n tosses of a fair coin (i.e., Y N Bin(n, %)). Recall that My(t) = 2\"(1 + e')\". For this Y, the condition in (c) is satised when t=log( a. ). na Provide two upper bounds on 1? (Y Z a) for a = 3%the probability that at least 3n/4 tosses give headsone using Markov's inequality and the other using the bound from parts (b) and (0). Which of the bounds is best as n > 00? Hint: It may be hetpit to note that 2\" = 6132)\" and.r log2