課程名稱︰機率導論
課程性質︰數學系必修課
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）︰2017/4/25
考試時限（分鐘）：110分鐘
試題 :
Introduction to Probability Midterm Examination 25 April 2017
1.(10 pts) An urn contains 30 red balls and 24 blue balls. They are withdrawn
one at a time without replacement until a total of 8 red balls have been
withdrawn. Find the probability that a total of k balls are drawn.
2.(10 pts) If there are 45 strangers in a room, what is the probability that
no two of them celebrate their birthday in the same day? (365 days a year)
3.(10 pts) Suppose that 5 percent of men and 0.25 percent of women are color
blind. A color-blind person is chosen at random. What is the probability of
this person being male? Assume that the population consists of twice as many
male as females.
4.(10 pts) A and B alternate rolling a pair of dice, stopping either when A
rolls the sum 10 or when B rolls the sum 8. Assuming that A rolls first,
find the probability that the final roll is made by A.
5.(15 pts) Suppose a random variable X has the Yule-Simons distribution:
4
P( X = n ) = ——————— , n = 1,2,....
n(n+1)(n+2)
(a) Show that the preceding is a probability mass function.
(b) Show that E(X) = 2.
(c) Show that Var(X) = ∞.
6.(15 pts) Suppose that random variable X has the Binomial(n,p) distribution.
Compare the Poisson approximation with the correct binomial probability.
(a) P( X = 2 ) when n = 8, p = 0.1 .
(b) P( X = 9 ) when n = 10, p = 0.95 .
(c) P( X = 0 ) when n = 20, p = 0.1 .
X
7.(15 pts) Find the probability density function of Y = e when X is normal
2
distributed with parameters μ and σ .
8.(15 pts) Verify that
α
Var(X) = ———
2
λ
when X is a gamma random variable with parameters α and λ.