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- 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.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?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.
- 6. Determine the constant c so that f(x)= cx, x-1, 2, 3,....10 satisfies the conditions of being a p. m. f. for a random variable X.V2,4. (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?
- 4. If the random variable X has an exponential distrib show that for a > 0, b>0, 1 f(x) = -x/², for x ≥ 0, e 2 P(X> a+b|x> a) = P(X > b).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)?2.3.7. Suppose X1 and X2 are discrete random variables which have the joint pmf p(x1, r2) = (3x1+r2)/24, (21, 2) = (1, 1), (1, 2), (2, 1), (2, 2), zero elsewhere. Find the conditional mean E(X2 1), when r1 = 1.