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Math 418/544 Practice Questions for Midterm Note: There is probably a greater emphasis here on more recent material as you have not had the opportunity to do as many questions here. This does not mean the same will be true for the actual midterm. 1. State carefully: (a) Jensen’s inequality. (b) The weak law of large numbers. 2. Give examples of the following (and justify your examples): (a) A probability density function f (x) such that if X has density function f then E(X) = 0 but the variance of X is infinite. (b) A pair of uncorrelated random variables which are not independent. 3/2 3. A random variable X with finite mean has characteristic function φ(t) = e−2|t| . (a) Find E(X). (b) Find the variance of X or explain why it doesn’t exist. 4. True or False. If True provide a proof. If False give (and justify) a counter-example (a) If F and G are each distribution functions of a random variable, then so is F (x)G(x). (b) If φ and ψ are each characteristic functions of a random variable, then so is φ(t)ψ(t). (c) If f is the probability density function of a non-negative random variable with mean 1, then g(x) = xf (x)1(x ≥ 0) is the probability density function of a random variable. L2 P (d) If {Xn } and X are random variables such that Xn → X, then Xn → X. 5. X is a random variable with a symmetric distribution and finite mean µ and variance σ 2 . Based on statistical data we are willing to assume that σ 2 ≤ 8. Based on this information find an upper bound on P (X ≥ 10). (You should find the best possible bound.) 6. George and Julia work at the campus coffee shop. The management wants to award a prize to the quicker worker. They will each be set the task of making 200 consecutive double decaf skim milk lattes and, for each, the total of the 200 independent times to accomplish this will be measured. If the two total times differ by more than 80 seconds, the prize will be awarded to the faster one, otherwise no prize will be awarded. The standard deviation of the time it takes each person to make a double decaf skim milk latte is 4 seconds. If the mean times for both George and Julia are actually the same, what is the approximate probability that George will get the prize? You can give your answer in terms of an integral. 7. Let H be the set of half spaces in Rd , that is, the set of sets H of the form H = {x ∈ Rd : u · x ≤ c}, where u ∈ Rd , c is a real number, and u · x is the scalar product. If P and Q are two probabilities on (Rd , B(Rd )) such that P (H) = Q(H) for all H ∈ H, prove that P = Q. Hint. Characteristic functions. 8. Assume X1 , X2 are i.i.d. random variables such that X1 and (X1 + X2 )/2 have the same distribution. (a) If E(X1 ) = µ is finite prove that X1 = µ a.s. (b) Give a counterexample showing that there are r.v’s X1 , X2 as above, and not a.s. constant r.v.’s, if we do not assume X1 has a finite mean.