15. This problem extends Problem 20.6. Let X, Y be random variables with finite mean. Show that (P(X ≤ x ≤ Y) - P(Y < x ≤ X))dx = E Y — E X.
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- 1. Random variable X has p.d.f -6z 6e fx(z) otherwise and let Y = eX. Calculate E(X)4. Let X, Y be independent random variables. Prove that E(XY)= E(X) E(Y) and hence show that V(X+Y) = V(X)+V(Y).7.4.3. X is the continuous uniform (0, 1) random variable. Given X = x, Y is conditionally a continuous uniform (0, 1+ x)random variable. What is the joint PDF fx,Y(x, u) of X and Y?
- If X₁, X2,..., Xn constitute a random sample of size n from an exponential population, show that X is a sufficient estimator of the parameter 0.2. Suppose one has n non-degenerate random variables X1, X2,, X, so that X1+ X2 + · · · + Xn = L for some constant L. (Recall that a random variable is non-degenerate if it is not a constant in disguise.) Show that there must be at least one pair of indices i j so that p(X;, X;) < 0.4. (a) Consider a nonnegative, integer valued random variable Y , and a random sample X1,., Xn. + Xy) in terms of E(X1) Assume Y and all the X;'s are independent. Find E(X1+ and E(Y). ... (b) Suppose each person who logs onto Amazon.com on Christmas Eve is expected to spend $80. One hundred people are randomly chosen to see how much is spent by them. Each of them visits Amazon.com with probability 60%. How much money do we expect this group of people to spend at Amazon.com?
- 3. If X and Y are independent uniform (0, 1) random variables, show that 2 (a + 1)(a + 2) E[|X - Yº] = for a>07.3.3. Random variables X and Y have joint PDF Be-(2=+3y) 120, y 2 0, fx x (z, y) – otherwise. Let A be the event that X + Y 3 1. Find the conditional PDF fx,YJA(X,).I roll a die with 10 faces numbered 0-9 and obtain a score X. What is E(X)? In order to obtain a random number between 0 and 99 I roll the die twice obtaining scores X and Y, then set Z = 10X + Y (i.e., the first roll determines the "tens", the second roll the "ones"). Is it true to say that E(Z) = 10E(X) + E(Y)?
- Suppose X is random variable whose p.d.f. is f2)=(2x-x²), 0, OSxs2 Suppose X is random variable whose p.d.f. is f(x)={4 elsewhere Find the mode , if it exists.4. Suppose X1, X2,..., X10,000 are independent random variables with E(X;) = 8 and Var(X;) = 4 for each i. Let Y = (X1+ X2 + ... +X10,000)/10,000. a. Find E(Y). b. Find Var(Y). c. What does the Chebyshev inequality say about P(|Y – E(Y)| 2 0.1)?7.4.8 Y = ZX where X is the Gaussian (0, 1) random variable and Z, independent of X, has PMF (1-p --1, Pz (2) - 2- 1. True or False: (a) Y and Z are independent. (b) Y and X are independent.