課程名稱︰機率導論 課程性質︰數學系大二必修 課程教師︰鄭明燕 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2016.04.12 考試時限（分鐘）：110 試題 : Introduction to Probability Midterm Examination 12 April 2016 1. (15 pts) Supposethat A and B are two events such that P(A∪B) = 4P(B), P(A∩B) = 0.2P(A∩B'), P(A'∩B') = 0.2 (a) Find the value of P(A) and P(B). (b) Find P(A|B). 2. (15 pts) A box contains 5 ballslabeled 1, 2, ..., 5. Three balls are selected from the box randomly and without replacement. Let X be the sum of the numbers. (a) What is the cumulative distribution function of X? (b) Find the probability that balls 1 and 3 are among the three selected balls. (c) Find the probability that ball 1 is selected given that ball 5 is selected. Are the events "ball 1 is selected" and "ball 5 is selected" independent? 3. (20 pts) Suppose that X is a discrete random variable with the probability mass function f (x) = cx^2, x = 1, 2, 3, 4, 5, for someconstant c. X (a) Find the value c. (b) Find the cumulative distribution function of X. (c) Find the pronability mass and cumulative distribution function of Y = -3X + 0.5. 4. (15 pts) Suppose that X is a Geometric random variable with the probability mass function f (x) = (1-p)^(x-1)p, x = 1, 2, 3, ...,and f (x) = 0 otherwise. X X (a) Find the expected value and variance of X. (b) Show that for any positive integers n and k, P(X = n+k|X＞n) = P(X = k). 5. (15 pts) Describe how to generate 100 observations from the Bata(1,β) distribution function whose density function is Γ(1+β) β-1 ──────(1-x) I (x), where β＞0. Γ(1)Γ(β) (1,0) 6. (10 pts) State and prove the Possion approximation to the Binomial distribution. 7. (10 pts) State the Normal approximation to the Binomial distribution. --

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