課程名稱︰機率導論 課程性質︰必修 課程教師︰陳 宏 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2013/06/20 考試時限（分鐘）：13:20PM-15:10 試題 : Introductory Probability 機率導論 期末考 考試時間: 13:20PM-15:10 Thursday, June 20th, 2013 1. (40 points) Suppose X and Y are independent exponential random variables both with parameter λ = 1. Find the probability that Y < 7X. 2. (50 points) The random variablee X is normally distributed with mean μ and variance σ^2. Moreover, P(X < 3) = 0.8413 and P(X < 5) = 0.9772. Determine μ and σ. 3. (50 points) We have three random variables N, Y, and X. It is known that N is the number of flips of a fair coin until the first head. Y is a uniform random variable taking real values on (0,N^2). Moreover, X is an exponential random variable with parameter √Y. (a) (15 points) Derive the probability mass function of N. (b) (15 points) Compute E[Y|N = 5] and E[Y]. (c) (20 points) Compute E[X|Y = 4] and E[X]. 提示: Use the conditional expectation formula. 4. (25 points) Let X and Y denote the values of two stocks at the end of a five-year period. X is uniformly distributed on the interval (0,12). Given X = x, Y is uniformly distributed on the interval (0,x). Determine Cov(X,Y). (Note that Cov(X,Y) = E[XY] - E[X]E[Y].) 5. (45 points) Let X_1, X_2, … be a sequence of independent and identically distributed random variables with distributtion function ╭ 0 x < 0 │ F_X(x) = ﹤ 1 - (1 - x)^2 0 ≦ x ≦ 1 │ ╰ 1 x > 1 and let V_n = max(X_1, X_2, … , X_n). (a) (15 points) Find P(V_n < c) for 0 < c < 1. (b) (15 points) Determine n such that P(V_n > 0.95) ≧ 0.9. (c) (15 points) Determine a such that, as n goes to infinity, the limit of P(V_n ≦ n^-a) exists and the limit is between 0 and 1. 6. (40 points) Let S_n = X_1 + … + X_n where X_1, …… , X_n be the i.i.d. Poisson random variables with λ = 1. (a) (25 points) Prove that S_n is a poisson random variable and determine P(S_n = n). (Note that the moment generating function of a Poisson random variable with λ is exp(λ(e^s - 1)).) (b) (15 points) Using the central limit theorem for a sum of Poisson random variables, compute n+√n lim e^-n Σ [(n^i)/i!]. n→∞ i=0 7. (50 points) Suppos that X_i are independent random variables which take the n values 2 and 0.5 each with probability 1/2. Let Y = Π X_i. i=1 (a) (15 points) Compute E[Y] (b) (35 points) 估計 P(Y > 1000) 當 n = 100 時. 提示: 當 Y > 1000 發生時，得至少有多少個X_i的取值為2. 8. (35 points) Let M_X(t) = c({[2+exp(-t)]/[2-exp(-t)]}^2) be the moment generating function associated with a discrete random variable X that takes integer values greater than or equal to -k, k > 0. (a) (10 points) Determine the constant c. (b) (10 points) Determine E[X]. (c) (15 points) Determine P(X = -1) and the value of k. 9. (40 points) Suppose the number of customers arriving in a store during a typical hour is Poisson distributed with parameter λ. Let p be the typical fraction of customers that are female. In other words, conditioning on the event that exactly n customers come during a given hour, the probability that k of them are female is C(n,k)p^k (1-p)^n-k. (a) (20 points) Show that the number of female customers arriving in a given hour is Poisson distributed with parameter pλ. (b) (20 points) Show that the event that no females arrive durring the hour is independent of the event that no male arrives during the hour. --

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