17. Suppose that X1, X2,..., Xn are random variables, such that E|xk| < ∞ for all k, and set Yn = max1

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.

2. Let X be a random variable with p.m.f., (1+ x? f(x) = -,x = -1,0,1,2,3 20 0 ,otherwise. Find E(3X2 + 6).

2. Let X and Y be random variables. Prove that |px,x| = 1 if and only if Y = a +bX for some a, b E R.

4. Let (, F, P) be a probability space and let X and Y be random variables on it. Assume that the random variable X takes on value 0 and 1, and the distribution of X is given by Px (0)=0.3, Px(1) = 0.7. Further assume that the random variable Y takes on value 0 and 1, and the distribution of Y is given by Py (0)=0.8, Py(1) = 0.2. (i) Assume that the random variables X and Y are independent. Find the joint distribution of X and Y, Px,y. [5 Marks] (ii) Give an example of a possible joint distribution Px,y such that the random variables X, Y are not independent. [15 Marks] Hint. Denote Px,y (0, 0) = = a, Px,y (0, 1) = b, Px,y(1, 0) = = C, Px,y (1, 1) = = d. We must have (why?) a, b, c, d = [0, 1], a+b+c+d= 1, a+b= 0.3, a+c=0.8. Choose a, b, c, d satisfying the above conditions and such that a 0.24.

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.

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)?

3. Let X be the random variable that takes on the integers {0, 1, 2, ..., 15} with equal probabilities. Define a new random variable Y = X + A, where A is a random variable that takes on the values {-1, 0, 1} with equal probabilities. If the RVs X and A are independent, find the mutual information between X and Y.

23. Suppose (X₁, ..., Xn} are random variables on the probability space (S, B, P) such that U i#j 1≤i, j≤n Define the relative rank Rn of Xn among (X₁, ..., Xn} to be Σ=1¹[X₁2Xn] P[ Ties]:= P Rn 1 17, Prove Rn is a random variable. [X₁ = X₁]} = 0. 1 on [ Ties ]º, on [Ties].

ii) Let X be a random variable taking three values: P(X= a₁) = P₁, P(X=a₂) = P2, P(X=a3) = P3, where p₁ + P2 + P3 = 1 and P₁, P2, P3 € (0, 1). Let A = {X = a₁} and G = {N, 0, A, Ac}. Prove that E (X³|G) = a³¹₁ + ª²P² + a²P3, P2 + P3 1A.