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.