課程名稱︰機率導論
課程性質︰數學系大二必修
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）︰2015/06/09
考試時限（分鐘）：60
試題 :
2
1. (25 points) Let X be a normal random variable with mean 0 and variance σ .
Let S be a random variable which is independent of X and such that P(S = 1) =
P(S = -1) = 1/2.
(a) (15 points) Show that SX is a normal random variable and determine its
mean and variance. (You can use moment generating function to show it.)
(b) (10 points) Show that X and SX are uncorrelated, but X and SX are not
independent.
2. (25 points) Let N, X , X be three independent random variables. Let Y =
1 2
N
Σ X where N is a random variable taking values 1 and 2 with probability 1/2
i=1 i
2
and 1/2 while E(X ) = μ and Var(X ) = σ . Determine E(Y) and Var(Y).
i i
3. (25 points) A basketball team scores baskets according to a Poisson process
with rate λ = 2 baskets per minute.
(a) (10 points) What is the expected amount of time until the team scores its
first basket?
(b) (5 points) Given that at the five minute mark of the game the team has
scored exactly one basket, what is the probability that the team scored
the basket in the first minute?
(c) (10 points) What is the probability that the team scores exactly three
baskets in the first five minute of the game? (Computation of the
numerical value is not necessary.)
4. (25 points) Show that a Poisson process is continuous in probability by
arguing
P(N - N > η) = P(N - N ≧ 1) < ε
t+h t t+h t
Hint: Associate the distribution of N - N and the distribution of N .
t+h t h
5. (25 points) Distribute n balls independently at random into n boxes. Let N
n
be the number of empty boxes. Denote N by the sum of I , ..., I , where I =
n 1 n i
1 if the i-th box is empty and 0, otherwise.
(a) (4 points) Write N in terms of I , ..., I .
n 1 n
(b) (6 points) Derive E(I ) and Var(I ).
1 1
(c) (10 points) Derive E(I I ) and Cov(I , I ).
1 2 1 2
-1
(d) (5 points) Show that N /n converges to e in probability as n goes to
n
infinity.
/*
1. 我修改了第4題的題敘，讓題意比較清楚。
2. 5(d)原本沒有"in probability"，這是考試中老師加上去的。
*/