Statistics 134, Fall 2002 Prof. PitmanFINAL EXAMName:SID number:SHOW CALCULATIONS, OR GIVE REASONS, ON ALL PARTS OF ALLQUESTIONS. DO NOT LEAVE NUMERICAL ANSWERS UNSIMPLIFIED.You may refer to your text and class materials.12345678910Total1. Each time a random number generator is run, it produces a pair of digits by makingtwo draws at random with replacement from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Thegenerator is run 5000 times. Let X be the number of times it produces the pair 00. Find aninteger n so that P (X ≥ n) is approximately 85%.2. A standard deck consists of fifty-two cards. Four of the cards are aces. Cards are dealtfrom the deck at random without replacement until two aces have appeared.Let X1be the number of cards dealt till the first ace appears. Let X2be the total numberof cards dealt till the second ace appears. So P (X2> X1) = 1.a) Find P (X1= 1, X2= 5).b) Find P (X1= 1 | X2= 5).3. I have three painted dice.• The first die has one red face and five green faces.• The second die has three red faces and three green faces.• The third die has five red faces and one green face.I pick one of the dice at random and roll it twice. Let R1be the event that the first rollshows a red face. Let R2be the event that the second roll shows a red face.a) Find P (R1).b) Are R1and R2independent?5. Let X and Y be independent normal variables, with E(X) = 10, SD(X) = 3,E(Y ) = 20, SD(Y ) = 5.Let V = Y − X + 2 and W = 4X − 2Y + 4.a) Find the correlation between V and W .b) Find E(V | W = 25).c) Find P (V > W + 20).4. Particles arrive at a Geiger counter according to a Poisson process with a rate of 2 perminute.a) Find the expectation and standard deviation of time between the arrival of the secondparticle and the arrival of the fifth particle.b) Find the chance that the fifth particle arrives more than two minutes after the secondparticle.6. Let X and Y have joint density given byf(x, y) =1211(x2+ xy + y2), 0 < x < 1, 0 < y < 10 otherwiseFind P (X < 1/4, X + Y < 1/2).7. A computer screen saver draws colored discs. Each disc is equally likely to be blue,green, yellow, or red, independently of all other discs. The radius of each disc (measured ininches) is chosen independently of the colors and radii of all other discs, according to thedensityf(r) =0.5r, 0 ≤ r ≤ 20 otherwisea) Let S be the area of the first disc drawn by the screen saver. Find the density of S.Recognize this as a named density and provide its parameters.b) Find the chance that the first four discs drawn by the screen saver all have areas biggerthan 10 square inches and are all of different colors.8. X and Y are independent and each is uniformly distributed on the interval (0, 1). Findthe c.d.f. of Y /X and sketch its graph.9. A survey organization is about to choose a random sample of 200 households. Supposethat for each i in the range 1 through 200, the number of residents in the ith sampledhousehold equals 1 + Ni, where N1, N2, . . . , N200are independent and identically distributedPoisson variables, each with parameter 2.Let X be the total number of residents in all the sampled households.a) Find the expectation and standard deviation of X.b) Find an approximate value of P (X > 650). Give a brief justification for your approx-imation.c) Find the exact distribution of X.10. A basket contains n balls labeled 1 to n. Near the basket there are n empty boxeslabeled 1 to n. Balls are placed in the boxes as follows. A ball is picked at random from thebasket and placed in Box 1. Then a ball is picked at random from the n − 1 balls remainingin the basket, and placed in Box 2. The process continues till all n balls have been placed.At the end of the process the basket is empty and each box contains exactly one ball.For each i from 1 to n, say that a match occurs at i if Ball i gets placed in Box i. Let Mbe the total number of matches.a) Show that E(M) = 1.b) Show that V ar(M) = 1.c) What is the approximate distribution of M when n is large? Give an intuitive justifi-cation, and show that your answer is consistent with the mean and variance of parts a)