Posted: April 6th, 2023

UCLA-MATH 170B Math 170b-Math 170B Practice Exam 2

Math 170B Practice Exam 21. (a) State Markov inequality.(b) Let X1 , · · · , Xn , · · · be i.i.d. random exponential variables with parameter λ = 1/2. And letN independent to X1 , · · · , Xn , · · · be a geometric random variable with parameter p = 0.4. LetY = X1 + · · · + XN Use Markov inequality to find an upper bound of P (Y ≥ 10).2. (a) State the strong law of large numbers.(b) Let X1 , · · · , Xn , · · · be a sequence of i.i.d. random variables taking values on [1, 3]. SupposeE[Xi ] = 2. Use weak law of large numbers to find a upper bound of the probabilityX1 + · · · + X1000 − 1000 ∗ 2> 0.1P10003. Give an example of sequence of random variables X1 , · · · , Xn , · · · such that E[|Xn − Xn−1 |] = 2for all n but Xn → 0 in probability. No explanation is necessary.4. In law of large numbers, we know that for X1 , · · · , Xn , · · · to be a sequence of i.i.d. randomvariables with E[Xi ] = µ for all i ≥ 1, Mn = (X1 + · · · + Xn )/n, and any > 0,lim P (|Mn − µ| ≥ ) = 0.n→∞So Mn is close to a constant. However, in central limit theorem, we also know we can have normalapproximation and approximate the cumulative distribution function of Mn using normal randomvariables. Will these two statements contradict with each other?15. Suppose a certain candidate is supported by 60% of all the voters. We randomly interviewn voters on whether they support the candidate. Let Mn be the percentage of the voters weinterviewed supporting the candidate. What is the number of voters we should interview, inorder to guarantee that Mn ∈ [59%, 61%] with a probability ≥ 0.95?2

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