Theorem 5.1 (Jensen's inequality) state without proof the Jensen's Ineg. Let X be a random variable, g a convex function, and suppose that X and g(X) are integrable. Then g(EX) < Eg(X).
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- Theorem 2.6 (The Minkowski inequality) Let p≥1. Suppose that X and Y are random variables, such that E|X|P <∞ and E|Y P <00. Then X+YpX+YpProve - If X and Y are independent random variables, then E(XY ) = E(X)E(Y ). (You can assume that both X and Y are either continuous or discrete random variables.)Theorem 7.6 (Etemadi's inequality) Let X1, X2, X, be independent random variables. Then, for all x > 0, P(max |S|>3x) ≤3 max P(S| > x). Isk≤n
- Theorem 1.6 (The Kolmogorov inequality) Let X1, X2, Xn be independent random variables with mean 0 and suppose that Var Xk 0, P(max Sk>x) ≤ Isk≤n Σ-Var X In particular, if X1, X2,..., X, are identically distributed, then P(max Sx) ≤ Isk≤n nVar X₁ x2Theorem 1.4 (Chebyshev's inequality) (i) Suppose that Var X x)≤- x > 0. 2 (ii) If X1, X2,..., X, are independent with mean 0 and finite variances, then Στη Var Xe P(|Sn| > x)≤ x > 0. (iii) If, in addition, X1, X2, Xn are identically distributed, then nVar Xi P(|Sn> x) ≤ x > 0. x²a) A median of a distribution of one random variable X of the discrete or continuous type is a value of X such that Pr (X 1/2 If there is only one such x, it is called the median of the distribution, Find the median of the following: f(x) = - o < X < 0 (1+x²)n b) We have continuous-type random variables X and Y having p.d.f given below: f(x, y) = 4xye-x²-y? -00 < X < o - 0 < Y < ∞ =0 else where Find marginal distribution of X and Y. c) We have discrete-type random variables X and Y having p.d.f given below: f(x, y) = x=1,2,3 y=1,2,3 4x+y Find Z=(3X-2Y) find mean and variance of Z.
- 4. At a particular gas station, gasoline is stocked in a bulk tank each week. Let random variable X denote the proportion of the tank's capacity that is stocked in a given week, and let y denote the proportion of the tank's capacity that is sold in the same week. The joint pdf of X and Y is given by f(x,y) = {³x (3x 0 if 0 ≤ y1. Entropy of functions of a random variable. Let X be a discrete random variable. Show that the entropy of a function of X is less than or equal to the entropy of X by justifying the following steps: H(X. g(x)) H(X) + H(g(X) | X) H(X); H(X,g(X)) H(g(x)) + H(X | g(X)) Thus H(g(X)) ≤ H(X). > H(g(X)). (2.1) (2.2) (2.3) (2.4)ACTIVITY 2. Consider the probability mass functions of the two investment options that are presented to a businessperson. Compute for the mean and variance of each investment. Investment A Profit P(x) x: P(x) x2 x2- P(x) (x) 10,000 10 3 5, 000 10 -3,000 10 Ex2. P(x) Investment B Profit P(x) x2 x* - P(x) x: P(x) (지) 2 20,000 10 5 16,000 10 3 -15,000 10 Compare the measures of variability for each investment. Mean Variance Standard Deviation Investment A Investment BSEE MORE QUESTIONS



