課程名稱︰機率 課程性質︰必修 課程教師︰劉長遠／林守德 開課學院：電資 開課系所︰資工 考試日期（年月日）︰2016/04/18 考試時限（分鐘）：180 試題 : Probability Midterm 2016/4/18 14:20~17:20 Prof: Cheng-Yuan Liou ***All work must be explained*** ***In grading the exam, considerably more weight will be given to the formulation of an expression (i.e. to setting up a fraction or setting up an integral) than to the evaluation of that expression (i.e. integral).*** 1. (10%) Mary and Tom park their cars in an empty parking lot that consists of N parking spaces in a row. Assume that each possible pair of parking locations is equally likely. Calculate the probability that the parking spaces they select are at most 2 apart (that is, at most one empty space between them). 2. (10%) Out of the students in a class, 60% are geniuses, 70% love chocolate, and 40% fall into both categories. Determine the probability that a randomly selected student is niether a genius nor a chocolate lover. 3. (40%) Assume that every gasoline filling station in Cambridge is open each day for a single, continuous, two-hour period. The opening times, t for all stations are identically distributed, mutually independent random variables with the PDF: ft(t0)↑ │ K │------┌──┐ (time at which any │ │ │ particular station opens) │ │ │ └───┴──┴─→t0 6 am. 10am. (a) (3%) Determine the numerical value of k. (b) (7%) If Oscar visits a randomly selected station at 9:30 a.m., what is the probability the station is open when he arrives? (c) Each day, Oscar visits his first station at 9:30 am. If it is closed, he tries another at 10:30 am. If the second station is also closed, he tries a third station at 11:30 am. If all three tries fail, he gives up for the day. (He needs one hour to go from any station to any other one. Note that, by 12:30 pm no stations are open.) (1) (10%) What is the probability Oscar was able to purchase gasoline on the forst day ge tried to do so? (2) (5%) Given that Oscar was able to purchase gasoline today, what is the probability he visited a total of exactly two gas stations today? (3) (10%) Given that Oscar bought gasoline at 11:30 am. Today, determine the conditional expected value and the conditional variance for today's opening time at the particular station at which he bought gasoline. (4) (5%) Let x be the number of days up to and including the first day on which Oscar is successful in his attempt to purchase gasoline. Determine the PMF Px(X0) 4. (15%) The compound PDF for random variables x and y is: f_x,y (x_0,y_0) = { Ax_0 If x_0 >= 0 and y_0 >= 0 and x_0 + y_0 <= 1 { 0 otherwise Let g = xy derive PDF f_g (g_0) 5. (15%) Random variables x and y are independent and are described by the PMF's: Px(x0)↑ Py(y0)↑ │ │ 1/2 │┌┐--┌┐ 2/3 │------┌┐ │││ ││ │ ││ │││ ││ 1/3 │┌┐ ││ └┴┴─┴┴→x0 └┴┴─┴┴→y0 0 1 1 7 Let x_1, x_2,......be independent experimental values of x. Let y_1, y_2,......be independent experimental values of y. Consider the following facts of these random variables μ = x_1 + x_2 + x_3 + x_4 + y_1 + y_2 + y_3 γ = 4x_1 + 3y_1 (a) (5%) How do you think the variance of μ and γ will compare? (b) (10%) Determine the Z-transforms and variances for random variables μ and γ 6. (10%) The country hospital is located at the center of a square whose sides are 3 miles wide. If an accident occurs within this square, then the hospital sends out an ambulance. The road network is rectangular, so the travel distance from the hospital whose coordinates are (0,0) to the point (x,y) is |x|+|y|. If an accident occurs at a point that is uniformly distributed in the square, find the expected travel distance of the ambulance. --

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