Due Friday February 17th before noon in the TA drop box, basement, AP&M. HOMEWORK 3 : HAND IN ONLY QUESTIONS: 2, 4, 8, 11, 13, 15, 21, 24, 27

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1 Exercise Sheet 3 Due Friday February 17th before noon in the TA drop box, basement, AP&M. HOMEWORK 3 : HAND IN ONLY QUESTIONS: 2, 4, 8, 11, 13, 15, 21, 24, A six-sided die is tossed. Determine the expectation of the following random variables: a. The value on the die. b. The square of the value on the die. c. The number of digits in the square of the value on the die. 2. Two fair six-sided dice are tossed independently. Let X be the value of the first die, and Y the value of the second. Determine the expectations of the following random variables. a. X + Y. b. X Y. 3. A fair coin is flipped n times independently. A k-run consists of a sequence of k consecutive heads that is not contained in a sequence of k + 1 consecutive heads. Let X k be the number of k-runs. Determine E(X k ). 4. Let X be a random variable with P (X = x) = 2 x for x = 1, 2,... and P (X = x) = 0 otherwise. Determine E(X). 5. Let X be a random variable and let µ = E(X). Show that E((X µ) 2 ) = E(X 2 ) µ Let X have the uniform pmf on {1, 2,..., n}. Determine a formula for E(X) as a function of n. 1

2 7. Let X have the uniform pdf on [0, 1]. Determine for each positive integer k the value of E(X k ). 8. Let X be a random variable whose cdf is given by { 1 1/x for x > 1 0 for x 1 Determine the expected value of X. 9. Let X be a random variable whose cdf is { 1 e x2 for x > 0 0 for x 0 Determine the expected value of X Let X be a continuous random variable whose cdf is F and such that F (0) = 0. Show that E(e X ) exists and E(e X ) = 0 e x F (x)dx. 11. Let X be a random variable whose cdf is given by x x + 1 for x 0 and 0 for x < 0. Determine a pmf for X and then the expectation of X if it exists. 12. Let X be a positive integer-valued random variable with cdf F. Show that if E(X) exists then E(X) = (1 F (x 1)). x=1 13. Cards are drawn one by one from a standard deck of 52 cards. Let X be the number of draws up to and including when the first diamond is drawn. If the cards are replaced in the deck, determine E(X). If the cards are not replaced in the deck, determine E(X). 2

3 14. A pair of fair coins are independently and simultaneously flipped, repeatedly. Determine the expected number of flips up to and including the first flip where the coins are both heads or both tails. 15. People are randomly selected one by one, and their eye-color recorded as blue or not blue. Suppose that 20 percent of people have blue eyes. Let X b be the number of people selected until the first person with blue eyes shows up, let X n be the number of people selected until the first person with non-blue eyes shows up, and let X be the number of people select until the first person with a different eye color to the preceding people shows up. Determine E(X b ), E(X n ) and E(X). 16. Determine the expected number of independent coin flips of a fair coin until the first two consecutive heads. 17. A fair coin is tossed 9 times independently. Determine exactly the following quantities. a. The probability that the number of tails is exactly two. b. The probability that the number of heads is at least 5. c. The probability that the number of heads is odd. d. The expected number of tosses until the first heads. e. The expected maximum of 10 and the number of tosses until the first heads. 18. Let X be a random variable with the binomial distribution Bin(n, p). Determine a. E(1/X). b. E(2 X ). 19. A gambler at a roulette wheel consistently bets one dollar on red, starting with a cash roll of one hundred dollars. The roulette wheel consists of 18 red slots, 18 black slots, and a 0 and 00. Determine the expected number of independent spins of the wheel until the gambler has no cash left. Is there a way to bet at least one dollar per turn so as to increase this expected number of spins? 20. Let X be a random variable with Poisson distribution Poi(1). Determine 3

4 a. E(X 2 ). b. E(2 X ). 21. Suppose that cars arrive at a traffic light according to a Poisson process with a rate of one car per minute starting from time t = 0 minutes. Determine a. The probability that no cars arrive in the first two minutes. b. The probability that no cars arrive ever. c. The probability that exactly one car arrives in the first minute. d. The probability that the second car arrives within 10 seconds of the first car. e. The probability that the second car takes more than two minutes to arrive. 22. Suppose n numbers are independently and uniformly picked from {1, 2,..., k}. Let X be the smallest number picked and let Y be the largest number picked. Determine the pmfs of X and of Y, and determine E(X) and E(Y ). 23. Let X 1, X 2,..., X n be independent real valued random variables. Determine the cdf of the minimum of X 1, X 2,..., X n. 24. A floor consists of parallel planks which have a two foot width. A circular disc of diameter one foot is randomly and uniformly dropped onto the planks. Determine the probability that the disc does not touch the line between two planks. Determine also the expected distance from the center of the disc to the closest line between two planks. 25* A floor consists of parallel planks which have a one foot width. A needle of length l is dropped onto the floor uniformly and randomly. Determine the probability that the needle does not touch the line between two planks. 26. A dealer has a pack of cards face down. Cards are drawn one by one, and revealed after you guess whether the card is red or black or pass. Score +1 for a correct guess, 1 for an incorrect guess, and 0 for a pass. Let X be the total score after the last card is revealed. 4

5 a. Develop a strategy that ensures X 1 no matter how the cards are ordered. b** Suppose the cards are in random order. Develop a strategy that maximizes E(X) and compute E(X) for this strategy. 27. A person repeatedly and independent flips a fair coin to decide whether to take a step to the left or to the right from a point X. Determine the expected number of times the person returns to X. 28* A person repeatedly and independently tosses a fair four-sided die to decide whether to take a step north, south, east or west from a point X. Determine the expected number of times the person returns to X. 5

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