STATISTICS 134 Practice FinalThere are 9 questions, worth a total of 49 p oints. Calculations should beworked through to an explicit numerical answer. Show your work!1. [5 points] Let U be a continuous r.v. with uniform distribution on (0, 1).Let X = logU1−U. Find a formula for the density function f(x) of X.2. [5 points] A box contains n tags numbered 1, 2, . . . , n. Two tags aredrawn without replacement, giving two numbers: write X for the smallerand Y for the larger number. Calculate P (Y = X + 1).3. [5 points] Consider Poisson random scatter with intensity λ on the plane.Let (X, Y ) be the coordinates of the random point of the scatter which isclosest to the origin. Find the joint density function f(x, y) of (X, Y ).4. [5 points] A roulette wheel has 38 slots, of which 18 are red and 18 areblack. In 100 spins of the wheel, let R be the number of “reds” and let B bethe number of “blacks”. Calculate the correlation cor(R, B).5. [5 points] A statistics class has 23 students. As part of an assignment,each student tosses a coin 200 times and records the number of heads. Whatis the chance than no student gets exactly 100 heads?6. [5 points] Let X and Y be independent r.v.’s with EX = EY = µ andvar X = var Y = σ2. Write Z = XY . Calculate var Z in terms of µ and σ.7. [8 points] Let X and Y be continuous r.v.’s with joint densityf(x, y) = 0.5 + 2xy if 0 < x < 1 and 0 < y < 1= 0 if not .(a) Find the marginal density of X.(b) Do X and Y have the same marginal density? E xplain.(c) Are X and Y independent? Explain.(d) Calculate P (X + Y < 1).18. [6 points] Let X1and X2be independent continuous r.v.’s with distri-bution functionF (x) = exp(−e−x), −∞ < x < ∞(a) What is the distribution function of X1+ c, for constant c ?(b) What is the distribution function of M = max(X1, X2) ?(c) True or false (and explain): for a certain constant c, the random variableX1+ c has the same distribution as the random variable M.9. [5 points] Let X and Y be independent r.v.’s with standard Normal(0, 1)distribution. Find the conditional distribution of X given X = Y
View Full Document