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STATISTICS 134 Practice Final There are 9 questions, worth a total of 49 points. Calculations should be worked through to an explicit numerical...

please, prac final for probability. need a solution!!

STATISTICS 134 Practice Final There are 9 questions, worth a total of 49 points. Calculations should be worked 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 = log U 1 - 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 are drawn without replacement, giving two numbers: write X for the smaller and 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 is closest 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 are black. In 100 spins of the wheel, let R be the number of “reds” and let B be the 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. What is 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 and var X = var Y = σ 2 . 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 density f ( x,y ) = 0 . 5 + 2 xy 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? Explain. (c) Are X and Y independent? Explain. (d) Calculate P ( X + Y < 1). 1
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8. [6 points] Let X 1 and X 2 be independent continuous r.v.’s with distri- bution function F ( x ) = exp( - e - x ) , -∞ < x < (a) What is the distribution function of X 1 + c , for constant c ? (b) What is the distribution function of M = max( X 1 ,X 2 ) ? (c) True or false (and explain): for a certain constant c , the random variable X 1 + 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 . 2
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