Name: MATH 360. Probablity Final Examination December 21, 2011 (2:00 pm - 5:00 pm) Instructions: The total score is 200 points. There are ten problems. Point values per problem are shown besides the questions. Show ALL your work and carefully justify your steps in your work! You may receive partial credit for partially completed problems. You are allowed to use a calculator and a two-sided sheet of notes for this exam. You may not use any other references or any texts. All cell phones, PDAs, ipods, laptops, etc. should be turned off and put out of sight. You may not discuss the exam with anyone but me. Please don t scratch on the line marked Score on the bottom of each page. My suggestion: You may read all questions before beginning and try to complete the ones you know best first. Good Luck!
(4points) 1. A package, say B 1, of 24 balls, contains 8 green, 8 white, and 8 purple balls. Another package, say B 2, of 24 balls, contains 6 green, 6 white, and 12 purple balls. One of the two packages is selected at random. Then, four balls are randomly selected without replacement from the contents of the selected package. Given that package B 1 was NOT selected, what is the probability of drawing 2 green balls, 1 white ball and 1 purple ball? (6points) (b) Find the probability that 2 green balls, 1 white ball and 1 purple ball are selected. (6points) (c) If the selected balls are 2 green, 1 white and 1 purple, find the probability that they were drawn from package B 2. Page 2 of 12
2. Suppose we have a random variable Y with the following cumulative distribution function: 0 y < 1, 1/8 1 y < 0, F (y) = P (Y y) = 3/8 0 y < 1, 4/8 1 y < 2, 1 y 2. (4points) Find the probability mass function p(y). (4points) (b) Find P (Y 1 Y 2). (6points) (c) What is the expected value and variance of Y? (4points) (d) What is the expected value and variance of 2Y + 5? Page 3 of 12
(5points) 3. Wires manufactured for use in a certain computer system are specified to have resistances between 0.12 and 0.14 ohms (the SI unit of electrical resistance). The actual measured resistances of the wires produced by Company A have a normal distribution with a mean of 0.13 ohms and a standard deviation of 0.005 ohms. What is the probability that a randomly selected wire from Company A s production will meet the specifications? (5points) (b) If four such wires are used in the system and all are from Company A, what is the probability that at least 3 of them will meet the specifications? (8points) (c) If 300 such wires from Company A are used, what is the (approximate) probability that at least 280 of them will meet the specifications? (4points) (d) What is the probability that the fourth wire used is the first one that does NOT meet the specifications? Page 4 of 12
4. Suppose that a company handles 10 jobs per day and the number of hours to complete each job, Y i, has an exponential distribution with parameter β = 3. Let T be the total time to complete all jobs in a day. Assume that Y i are independent. (7points) What is the distribution of T = 10 i=1 Y i? Explain. Also find E(T ). (3points) Now suppose that the company has determined that the number of jobs per day, N, in fact varies from day to day with an average of 10 jobs per day. (b) What is a reasonable choice for the distribution of N? Be sure to specify the value(s) of the parameter(s). (Binomial/Geometric/Hypergeometric/Poisson/Normal/Gamma) (5points) (c) What is the probability that there are exactly 20 jobs in TWO days? Page 5 of 12
Note that T = N i=1 Y i is now the sum of a random number of random variables. (3points) (d) E(T N = n) =? (5points) (e) E(T ), the expected total time to complete all jobs? (7points) (f) V (T ), the variance of the total time to complete all jobs? Page 6 of 12
5. Let Y 1 and Y 2 be random variables with the following joint probability density function: { k(1 y 2 ) 0 y 1 y 2 1, f(y 1, y 2 ) = 0 elsewhere. (5points) Find the value of k that makes this a joint probability density function. (8points) (b) Find the marginal probability density function of Y 1 and Y 2. (5points) (c) Find the conditional probability density function for Y 2 given Y 1 = y 1. Page 7 of 12
(6points) 6. Recall Weak Law of Large Numbers (WLLN): Let Y 1,, Y n be i.i.d. r.v.s with E(Y i ) = µ < and V (Y i ) = σ 2 <, i = 1,, n. Then, Ȳ = 1 n (Y 1 + +Y n ) converges in probability toward µ, as n. Show that E(Ȳ ) = µ and V (Ȳ ) = σ2 n. (4points) (b) State Tchebysheff s Theorem. (10points) (c) Use above to prove WLLN. Page 8 of 12
(3points) 7. Let the c.d.f. of a random variable Y be F Y (y) = y 2, 0 y 1; 0, y < 0; and 1, y > 1. Is Y continuous or discrete? Explain. (10points) (b) Let U F Y (Y ) = Y 2. Use the method of distribution functions to find the probability density function of U. (3points) (c) Does U have the same distribution as Y? Page 9 of 12
8. A bottle machine can be regulated so that the amount of fill dispensed by the machine per bottle, Y is normally distributed with mean µ ounce and standard deviation σ ounce. A random sample of n filled bottles is selected from the output of the machine on a given day, and the ounces of fill, Y 1,..., Y n, are measured for each. Let Ȳ = 1 n (Y 1 + + Y n ). (10points) σ2 Given Ȳ N(µ, n ), use the method of moment generating functions to further find the distribution of Z = Ȳ µ σ/ n. Recall the definition of a t distribution: Let Z be a standard normal r.v. and W be a χ 2 - Z distributed r.v. with ν d.f. Then, if Z and W are independent, we say that T = has a t distribution with ν d.f. (10points) (b) Show that E(T ) = 0 and V (T ) = ν ν 2, for ν > 2. (W/ν) Page 10 of 12
(4points) (c) What is the distribution of Ȳ µ S/ n (Ȳ µ)/ σ = n (n 1)S 2 σ 2 /(n 1)? Explain. (6points) (d) Suppose the population mean µ and the population variance σ 2 are unknown and n = 8. Find the approximate probability that Ȳ will be within 3S/ n of the true population mean µ. (5points) 9. In a playlist of 10 songs, 3 songs are from the band Queen, 4 songs are from the band Poison, and 3 songs are from Lady Gaga. The playlist is shuffled and the first three songs are played. Let Y denote the number of Lady Gaga songs played in the first three songs played. What is the probability distribution of Y? Be sure to specify the value(s) of the parameter(s). (5points) (b) Find the mean and variance for Y. Page 11 of 12
(8points) 10. Answer the following questions. Suppose the times that a cashier spends processing individual customer s order are independent random variables with mean 2.5 minutes and standard deviation 2 minutes. What is the approximate probability that it will take more than 4 hours to process the orders of 100 people? (6points) (b) TRUE or FALSE: Suppose two random variables Y 1 and Y 2 are independent. Then Cov(Y 1, Y 2 ) = 0. (Justification required) (3points) (c) TRUE or FALSE: Suppose two random variables Y 1 and Y 2 have bivariate normal distribution. Then zero correlation coefficient between two random variables Y 1 and Y 2 implies that they are independent. (No justification required) (3points) (d) TRUE or FALSE: Let Y 1 and Y 2 have joint density function { e (y 1+y 2 ) y 1 > 0, y 2 > 0 f(y 1, y 2 ) = 0 otherwise Then, Y 1 and Y 2 are independent. (No justification required.) Page 12 of 12