Question: 35. (a) Prove that ????[????] = ????[????|???? < ????]????(???? < ????) + ????[????|???? = ????]????(???? = ????) HINT: Define an appropriate random variable and then

35.

(a) Prove that

????[????] = ????[????|???? < ????]????(???? < ????) + ????[????|???? = ????]????(???? = ????)

HINT: Define an appropriate random variable and then compute ????[????] by conditioning on it.

(b) Use part

(a) to prove Markov’s inequality, which states that if

????(???? = 0) = 1, then for ???? > 0,

????(???? ≥ ????) ≤

????[????]

????

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