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Math 302 Final Exam
Section: 101 Instructor: Ed Perkins Duration: 2.5 hours.
Instructions:
- Write your name and student ID on every page.
- This examination contains eight questions worth a total of 102 points.
- Write each answer very clearly below the corresponding question (Use back of page if needed). Simplify your answer as much as possible but answers may include factorials, “choose” symbols or the exponential function. You may also use the function Φ(a) = √^1 2 π
∫ (^) a −∞ e
−x^2 / (^2) dx in your answer.
- No calculators, books, notebooks or any other written materials are allowed.
- Good luck!
(10 points) 1. (a) Define carefully: The cumulative distribution function of a random variable X.
(b) A random variable X has cumulative distribution function
F (x) =
0 if x ≤ 0 x 2 if 0^ < x^ ≤^1 1 2 +^
1 3 (x^ −^ 1)^ if 1^ < x^ ≤^2 5 6 +^
1 6 (x^ −^ 2)^ if 2^ < x^ ≤^3 1 if x > 3.
(i) Find P ( 12 < X ≤ 32 ).
(ii) Find P (X = 1).
(iii) Find the probability density function of X.
(15 points) 3. (a) (i) Define the correlation coefficient of two random variables X and Y.
(ii) Give an example of two random variables with correlation coefficient −1.
(b) When flying from Vancouver to Seattle a passenger must first check in with their airline, then pass security and then clear U.S. customs and immigration. These 3 procedures take random times T 1 , T 2 and T 3 , respectively, with unknown distribu- tions. Past data suggests that their mean values are 15 minutes, 20 minutes and 15 minutes, respectively, and their variances are are 36, 100 and 64, respectively. Moreover, the time for check-in, T 1 , is uncorrelated with the time to pass security and also with the time to clear customs, but T 2 and T 3 have correlation equal to 1 /2. (i) Find the mean and variance of the total time T for a passenger to complete the three procedures.
(ii) Using only the given information, find an upper bound for the probability that T is greater than 90 minutes.
(12 points) 4. (a) A and B are events such that P (A) = 0.5, P (B) = 0.6 and P (Ac^ ∩ Bc) = 0.1. Find P (A ∩ B) and P (A ∪ B).
(b) Let X be a geometric r.v. with parameter 1/3 and Y a r.v. receiving the values 1 , 2 or 3 with probability 1/3 each. If X and Y are independent, compute the probability P (X + Y = 4).
(11 points) 6. A key chain holds N different keys (N ≥ 2) and only one of them opens the door to your apartment. You try the keys on the key chain at random until the door unlocks. Let X be the number of times you tried unlocking the door. What is the probability mass function of X and the mean value of X when:
(a) After trying a key you leave it on the key chain.
(b) After trying a key you remove it from the key chain.
(11 points) 7. Let Z 1 and Z 2 be independent standard normal random variables. Find the following:
(a) E[e^3 Z^1 +2Z^2 ].
(b) The probability density function of e^3 Z^1 +2Z^2.
