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+Yp
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- 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≤n2. Let X be a random variable. (a) Show that, if E X2 = 1 and E X4details are in the imageTheorem 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 x23A file consisting of X packets is sent over the network to an end user2. Let the independent random variables X1 and X2 have Bin(0.1,2) and Bin(0.5, 3), respectively. (a) Find P(X1 = 2 and X2 = 2). (b) Find P(X1 + X2 = 1). (c) Find E(X1 + X2). (d) Find Var(X1 + X2).Suppose that X is a geometric random variable with parameter π. Show that P(X = n+k|X > n) = P(X = k).Let A, B, and C be independent random variables, uniformly distributed over [0,3], [0,2], and [0,4] respectively. What is the probability that both roots of the equation Ax2Bx+C = 0 are real?A bucket has three marbles. Two are black and one is white. You keep drawing until you get a while marble. Every time you draw a black marble you put it back into the bucket and also add a new white marble to the bucket. The random variable X denotes the total number of marbles you have drawn from the bucket. Find P(X = k) as a function of k. Compute E(X).show work, need it to studyRandom variable X has PMF PX(x) = a2 x for x = 1, 2, 3, 4; and PX(x) = 0 otherwise. Random variable X has PMF PX(x) = a2 x for x = 1, 2, 3, 4; and PX(x) = 0 otherwise.SEE MORE QUESTIONS