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□ende ^r， ^ , P, : >Y)< le、p(- ： 入 ).

Question Let Xi, X» ... be ¹ ncorrelated 丄va ⁵ ables /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, S _n /n — z/ _n 0 in L ² and in probability. 'Not。that 厂，^rgen。 厂 in A、me 二亠 … ^L(S _n /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(M _n )).

Define I to be the true integral J； fdx. Show that I _n I in probability. (2) Use Chebyshev's inequality to estimate P(|I _n — I )| > a/n ¹ / ² ).