1 Basic continuous random variable problems

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1 Name M362K Final Here are problems concerning material from Chapters 5 and 6. To review the other chapters, look over previous practice sheets for the two exams, previous quizzes, previous homeworks and work out problems in the book that weren t on the homework. Contents 1 Basic continuous random variable problems 1 2 Expected value 1 3 Computational problems 2 4 Independent random variables 3 5 Marginal and conditional densities 3 6 Covariance/Variance/Expected Value 4 7 Chebyshev/Markov 5 8 Normal random variables 5 9 Normal approximations/ central limit theorem 6 1 Basic continuous random variable problems 1. The lifetime of certain battery is an exponential random variable X with mean 1000 (in hours). Given that the battery has already lasted 1000 hours, what is the probability that it will last another 1000 hours? In other words, P (X 2000 X 1000) =? Hint: the density function of X is { 1 f X (x) = 1000 e x/1000 if x 0 2. Let X be a random variable with probability density function Find c. Then find P ( 2 < X < 1). f(x) = ce x, < x <. 2 Expected value 3. A certain road is 100 miles long from point A to point B. There are bus repair stations at points A, B and C which is located 30 miles from point A. If the next accident occurs at a point that is uniformly distributed along the length of the road, what is the expected distance from the accident to the closest repair station?

2 4. You and your friend agree to meet at a coffee shop around noon. Suppose your arrivial time (in minutes after noon) is exponential with mean 10 and your friends is also exponential with mean 10. Assuming the two arrival times are independent, how much time will you wait on average for your friend? It is OK to set up the integral without evaluating. 3 Computational problems 5. Let X be a continuous random variable with density { 1/x 1 < x < e f(x) = (a) Find the distribution function of X. (b) P (X > 2) =? (c) E[X] =? (d) E[X 2 ] =? (e) Find the distribution function of Y = X 2 (f) Find the density function of Y. 6. Let X, Y be random variables with joint density function { cxy 0 x y 2 f(x, y) = where c > 0 is a constant. In the questions below, it is OK to leave your answer in integral form without evaluating it. (a) Find c. (b) Find P (X + Y 2). (c) Find E[X]. (d) Find the density of Y. (e) Find f X (x Y = 1), the density of X given that Y = 1. (f) Are X and Y independent? 7. Let X, Y be random variables with joint density function { c(x + y) 0 x 1, 0 y 1 f(x, y) = where c > 0 is a constant. Compute the following. You can leave your answers in integral form. (a) c =? (b) P (X + Y 1)

3 4 Independent random variables 8. Let X, Y be exponential random variables with parameters λ 1, λ 2. Suppose X and Y are independent. (a) Compute the density function of X + Y. (b) P (X + Y 1) =? 9. Let X and Y be uniformly distributed over the interval [0, 1]. Compute the density function of X Y assuming X and Y are independent. Hint: find the distribution function of X Y first. 10. Let X, Y be independent exponential random variables such that E[X] = 10 and E[Y ] = 20. (a) Find the joint density function of (X, Y ). (b) Find P (X 5, Y 10). 11. Suppose X, Y, Z are uniformly distributed over (0, 1). Suppose they are also independent. (a) Find P (X 2 + Y 2 Z). (b) Compute P (X Y +Z). (It is OK to just set up the integral without computing). 12. You have two friends, Alice and Bob and they both promised to call you sometime after 7pm on Saturday. So you are sitting at home at 7pm on Saturday and you decide to go out with either Alice or Bob, whomever calls you first. Let A be the amount of time before Alice calls you, and B the amount of time before Bob calls you. Assume A and B are independent exponential random variables; the expectation of A is 60 minutes and the expectation of B is 30 minutes. What is the probability that you will go out with Alice? 5 Marginal and conditional densities 13. Suppose X, Y are random variables with joint density function { 4xy 0 x, y 1 f X,Y (x, y) = (a) Find the density of X. (b) Are X and Y independent? 14. Let (X, Y ) be uniformly distributed over the circle of radius 10 with center at (0, 0). (a) Compute the density of X. (b) Compute f X (x Y = 6), the density of X conditioned on Y = 6.

4 6 Covariance/Variance/Expected Value 15. Suppose that every day, the price of a certain stock either increases by 30% or decreases by 10% and that the probability that it increases is 1/2. If it starts out at 100, after n days what is the expected value of the stock? You can assume the different days are independent. 16. Suppose X, Y, Z are pairwise uncorrelated (this means Cov(X, Y) = Cov(Y, Z) = Cov(X, Z) = 0) E[X] = E[Y ] = E[Z] = 0 and Var(X) = Var(Y) = Var(Z) = 1. Find Cov(X + Y, Z Y ). 17. Ten hunters are shooting at 20 different ducks. Each hunter aims at a duck at random and hits it with probability 1/2 independently of the other hunters. (a) Let X be the number of hunters that hit a duck. What is E[X]? What is Var(X)? (b) What is the probability that the first hunter does not hit the first duck? (This means that either he does not aim at the first duck or he aims and misses). (c) Let Y be the number of ducks hit. What is E[Y ]? What is Var(Y)? 18. Suppose n people go to a dinner party. Each person checks in their hat. The host accidentally mixes up the hats and, in a vain attempt to avoid embarassment, hands them back to guests at random. Let X be the number of people that receive his/her hat back. Find the variance of X. (Hint: again you can write X = X X n for a suitable choice of X i s). 19. Let (X, Y ) be uniformly distributed over the triangle with vertices ( 1, 0), (1, 0), (0, 1). Compute Cov(X, Y). Are X and Y positively correlated, negatively correlated or uncorrelated? Are X and Y independent? 20. Let (X, Y ) be uniformly distributed over the triangle with vertices (0, 0), (1, 0), (0, 1). Compute Cov(X, Y). Are X and Y positively correlated, negatively correlated or uncorrelated? Are X and Y independent? 21. Let (X, Y ) be uniformly distributed over the triangle with vertices (0, 0), (1, 0), (0, 2). (a) Compute Cov(X, Y). (You can leave your answer in terms of integrals without evaluating) (b) Are X and Y positively correlated, negatively correlated or uncorrelated? 22. Roll two dice. Let X be the number on the first die and Y be the number of the second die. (a) Compute Cov(X + Y, X Y ). (b) Are X + Y and X Y independent? Explain!

5 23. Suppose an urn contains 30 red balls, and 70 blue balls. You pick 20 balls at random without replacement. Let X be the number of red balls chosen. Compute the variance of X. 24. Suppose an urn contains 30 red balls, 40 blue balls and 50 yellow balls (for a total of 120). You pick 12 balls at random. Let X be the number of red balls chosen. For the questions below, compute the answer exactly without any long summations. (a) Assume that the balls are chosen with replacement. Find E[X]. (b) Assume that the balls are chosen without replacement. Find E[X]. 25. (Balls and Boxes) Suppose there are n balls and n boxes. Each ball is placed in a box at random. Each box can contain an unlimited number of balls. Moreover the function assigning to balls to boxes is equally likely to be any of the n n functions. Find the expected value and the variance of the number of empty boxes. 7 Chebyshev/Markov 26. If a post office handles, on average, 10,000 letters a day, then provide an upper bound on the probability that they will handle more than 20,000 letters tomorrow. Use Markov s inequality. 27. If the number of letters a post office handles in any given day is a random variable with mean 10,000 and variance 2,000 then use Chebyshev s inequality to provide an upper bound on the probability that they will handle more than 20,000 letters tomorrow 28. Suppose X is a non-negative random variable and P (X 100) = 1/2. What is the smallest possible value of E[X]? 8 Normal random variables 29. Suppose that X 1, X 2,... are i.i.d. normal random variables with mean 0, variance 1 and S n = n i=1 X i. Determine the smallest value of n so that P ( Sn 1) n Hint: if χ is the standard normal then P ( χ 2) = You and your friend agree to meet at noon. Your arrival time in minutes after noon is normally distributed with mean 0 and standard deviation 3. Your friend s arrival time in minutes after noon is normally distributed with mean 1 and standard deviation 4. Assume that the two arrival times are independent. What is the probability that you will have to wait at least 4 minutes? 31. The distributions of the grades of the students of probability and calculus at a certain university are normally distributed with parameters µ = 65, σ 2 = 418 and µ = 72, σ 2 = 448 respectively. Dr. Olwell teaches a probability section with 22 students and a calculus section with 28 students. What is the probability that the difference between the averages of the final grades of these two classes is at least 2? You may assume independence. Write your answer in terms of the function Φ(t) = 1 t /2 2π e x2 dx.

6 32. A weighted coin lands on heads with probability 0.8. Suppose it is flipped 100 times and let X denote the number of times it lands on heads. (a) What is E[X]? (b) What is Var(X)? (c) Use normal random variables to approximate P (X 90). Leave your answer in terms of the Φ-function numbers are selected independently and uniformly from the interval [0, 1]. Use the central limit theorem to approximate the probability that the average of these numbers is within 0.01 of 1/2. Hint: the standard deviation of the random variable X that is 1 uniformly distributed in [0, 1] is 12. Leave your answer in terms of the Φ-function. 34. Toss a fair coin twice. You win $1 if at least one of the two tosses comes out heads. You neither win nor lose anything otherwise. (a) Assume that you play this game 300 times. What is, approximately, the probability that you win at least $250? (Use the normal approximation). (b) Approximately how many times do you need to play so that you win at least $250 with probability at least 0.99? 9 Normal approximations/ central limit theorem numbers are selected independently and uniformly from the interval [ 1/2, 1/2]. Use the normal approximation to approximate the probability that the average of these numbers lies in the interval ( 0.1, 0.1). You can leave your answer in terms of the Φ- function. 36. For a survey, we contact N people at random and ask them whether they plan to vote for candidate A. How large should N be so that there is at least a 99% chance that the sample mean determined by the survey is within 10% of the true mean? Use the normal approximation. 37. Suppose we contact 10,000 people at random and ask them whether they plan to vote for candidate A. Suppose that 60% of them say yes. Give a 95% confidence interval for the true percentage of people that plan to vote for A. 38. A fair die is rolled 100 times. Use the central limit theorem to approximate the probability that the sum of the outcomes is more than 400. Leave your answer in terms of the Φ-function. You do not have to simplify. Hint: if X is the value of a single die roll then E[X] = 3.5 = 7 35 and Var(X) = An astronomer is trying to estimate the distance from her observatory to a certain star. She makes 100 different measurements of this distance and takes the average of these 100 measurements to be her estimate. Suppose that the error in each measurement is a continuous random variable with mean 0 and variance σ 2 = 4 light years. Suppose also that the measurements are independent. Use the Central Limit Theorem to estimate

7 the probability that her estimate of the distance is less than the true distance by at least 1 light-year. Leave your answer in terms of the Φ-function. Hint: if X i is the error in the i-th measurement, then difference between her estimate and the true error is the mean Y = (X X 100 )/100. The question asks P (Y 1) (which is the same thing as P (Y 1)). 40. Suppose that 40% of all voters in this county are Republicans. If 100 are selected at random what is the probability that over 50 Republicans are selected? Use normal random variables to obtain an approximation. Leave your answers in terms of the Φ-function. 41. A fair die is rolled = 420 times. Use the central limit theorem to approximate the probability that the sum of the outcomes is between 65 and 75. Leave your answer in terms of the Φ-function. Hint: if X denote the value of a single die roll then Ē[X] = 7 2 and V ar(x) =

8 Some helpful formulae (this will be on the exam; no need to memorize it) Type of Random Variable Density or Mass Function Expected Value Variance Binomial(n, p) P (X = k) = ( n k) p k (1 p) n k (0 k n) np np(1 p) λ λk Poisson(λ) P (X = k) = e (k N) λ λ k! Geometric(p) P (X = k) = (1 p) k 1 1 p p, (k N) 1/p p 2 Exponential(λ) f(x) = λe λx 1 1, (x 0) λ λ 2 Normal(µ, σ) f(x) = 1 /2σ 2 2πσ 2 e (x µ)2 µ σ 2 Uniform over an interval [a, b] Var(X) = E[(X µ) 2 ] = E[X 2 ] E[X] 2. f(x) = b a a+b (x [a, b]) 2 2 Cov(X, Y ) = E[(X E[X])(Y E[Y ])] = E[XY ] E[X]E[Y ]. (Markov s inequality) P (X 0) P (X t) E[X]/t. (Chebyshev s inequality) P ( X µ t) Var(X) t 2. (b a) 2 12