EE 5110: Probability Foundations Tutorial 7 July-November 2015 1. [Properties of Simple functions] Let (Ω, F, µ) be a measure space and φ be a non-negative simple function. Then, prove the following: R R R (a) If C and D are disjoint measurable sets, then φdµ = φdµ + φdµ. C∪D C D R (b) The function υ : F → [0, ∞] defined by υ(A) = φdµ, for A ∈ F, is a measure on Ω. A R R (c) If A and B are measurable sets such that A ⊆ B and µ(B \ A) = 0, then φdµ = φdµ. B A 2. [Properties of Non-negative measurable functions] Let (Ω, F, µ) be a measure space. (a) Let fn be a sequence of non-negative measurable functions. Then, prove that for every n ∈ N, n n R ∞ ∞ R R P R P P P fk dµ = fk dµ. And if f = fk , then prove that f is measurable and f dµ = fn dµ. n=1 k=1 k=1 k=1 R (b) Let g be a non-negative measurable function and υ : F → [0, ∞] defined by υ(A) = gdµ, A ∈ F. A Show that R R υ is a measure on Ω. Further, prove that for any non-negative measurable function h, h dυ = hg dµ. 3. [Integration and Expectation] (i) Consider the measure space (R, B(R), λ), and define f : R → R. Find out the Lebesgue integral of the function f for the following cases, (a) f (ω) = ω, for ω = 0, 1, .., n 0, elsewhere. (b) 1, for ω = Qc ∩ [0, 1] 0, elsewhere. n, for ω = Qc ∩ [0, n] 0, elsewhere. f (ω) = (c) f (ω) = (ii) Consider the probability space (Ω, F, P), and define the random variable X : Ω → R. Find E[X] for the following cases, (a) Ω = {ω1 , ω2 , .., ωn }, with P(ωi ) = 1/n for i = 1, 2, .., n and X = IA , where A = {ω1 , ω2 , .., ωm } with 1 ≤ m ≤ n. (b) In part (a) if X is defined as follows, i, for ωi ∈ A X(ωi ) = 0, elsewhere. 4. [Jensen’s Inequality] (a) A function g:R → R is said to be convex if for all real y there exists a λ depending on y such that g(x) ≥ g(y) + λ(x−y), ∀x. Show that if g is convex and X is a random variable, then E[g(X)] ≥ g(E[X]). (b) Suppose m is a positive integer, and X is a random variable taking values x1 , x2 , ..., xN ≥ 0 with equal N P probabilities, and the sum x1 + ... + xN = 1. Is it true that N 1−m ≤ xm i ? i=1 1 5. [Impulse and Impulse Train] Consider the Real line endowed with the Borel σ-algebra, and let c ∈ R be fixed. Then the Dirac measure at c, denoted as δc , is defined on (R, B(R)) as follows. For any Borel set A, δc (A) = 1 if c ∈ A, and δc (A) = 0 if c ∈ / A. It is quite easy to see that (R, B(R), δc ) is a measure space (indeed, it is a probability space). The Dirac measure is referred to as unit impulse in the engineering literature, and sometimes (incorrectly) called a Dirac delta “function”. R (a) Let g be a non-negative, measurable function. Show that g dδc = g(c). Now, let us define a counting measure on (R, B(R)) as µ(A) = ∞ P δn (A). In words, µ(A)simply counts the n=1 number of natural numbers contained in the Borel set A. In engineering parlance, the counting measure is called an impulse train. (b) Let g be a non-negative, measurable function. Show that R g dδc = ∞ P g(n). Thus, summation is just a n=1 special case of integration. In particular, summation is nothing but integral with respect to the counting measure! 6. [Property of Expectation] Let X be a non-negative random variable (not necessarily discrete or continuous) with E[X] < ∞. (a) Prove that lim nP(X > n) = 0. [Hint: Write E[X] = E[XI{X≤n} ] + E[XI{X>n} ].] n→∞ (b) Prove that E[X] = R∞ P(X > x) dx. Yes, the integral on the right is just a plain old Riemann integral! R [Hint: Write out E[X] = x dPX as the limit of a sum, and use part (a) for the last term.] 0 We say a random variable X is stochastically larger than a random variable Y , and denote by X ≥st Y , if P(X > a) ≥ P(Y > a) ∀a ∈ R. (c) For non-negative random variables X and Y , show that if X ≥st Y , then E[X] ≥ E[Y ]. 7. [Expectation of Discrete RV’s] (a) On a given day, your golf score takes values from the range 101 to 110, with probability 0.1, independent of other days. Determined to improve your score, you decide to play on three different days and declare as your score the minimum X of the scores X1 , X2 and X3 on the different days. By how much has your expected score improved as a result of playing on three days? (b) A stock market trader buys 100 shares of stock A and 200 shares of stock B. Let X and Y be the price changes of A and B, respectively, over a certain time period, and assume that the joint PMF of X and Y is uniform over the set of integers x and y satisfying −2 ≤ x ≤ 4 and −1 ≤ y − x ≤ 1. Compute the trader’s expected profit. 2