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.

16. Show that, if X and Y are independent random variables, such that E|X|< ∞, and B is an arbitrary Borel set, then EXI{Y B} = EX P(YE B).

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).

3. Let X1, X2,..., X, be independent, Exp(1)-distributed random variables, and set V₁₁ = max Xk and W₁ = X₁+x+x+ Isk≤n

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.

6. Let, for p = (0, 1), and xe R. X be a random variable defined as follows: P(X=-x) = P(X = x)=p. P(X=0)= 1-2p. Show that there is equality in Chebyshev's inequality for X. This means that Chebyshev's inequality, in spite of being rather crude, cannot be improved without additional assumptions.

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)?