IOE516 - Stochastic Processes II - Problem Set 1 Winter 2023
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IOE516 - Stochastic Processes II - Problem Set 1
Winter 2023
Instruction:
• Due date: On Canvas
• Format: PDF submission to CANVAS
Question (Chebyshev's other inequality.) Let 扌:股 T 股 and g :股 T 股 be bounded and
increasing functions. Prove that, for any r.v. X,
.(X)g(X)) > E(扌(X))E(g(X))
Question Let X have Poisson (A) distribution and let Y I ave Poisso (2)、g strib tion. (i) Prove P(X > y) < exp( —(3 —必)入)if X and Y are independent, ii) Fmd co st ants 1 < o, c > 0, not depending on 入,such that, without assuming …丄□ender,^ , P, : >Y)< le、p(- :入).
Question Let Xi, X» ... be 1 ncorrelated 丄va 5ables /ith「(Xi)= 四 and var(X"/今 — 0
as 今 T 8. Let & = Xi + ... + *九 a丄丄二 几=丄(5^) " si)w t^at a_ n — oc, Sn/n — z/n 0 in L2 and in probability. 'Not。that 厂,^rgen。厂 in A、me 二亠…^L(Sn/n — z/n)2 T 0 as tz T oc.)
Question (M nt< Carlo Inf 聖;"Ion): (1) Let / be a measurable function on [0,1] with J0 |/(x)^dx oc. Let :丄i, Mr, . . oe independent and uniformly distributed on [0,1], and let
In = nT(KMi) + ...f(Mn)).
Define I to be the true integral J; fdx. Show that In I in probability. (2) Use Chebyshev's inequality to estimate P(|In — I)| > a/n1/2).
Question Suppose events An satisfy P(An) — 0 and
oo
E P(A" An+i) < 8.
n=1
Prove that P(An occurs infinitely often) = 0.
2023-02-02