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）︰2014/6/19
考試時限（分鐘）：13：20～15：20
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. Suppose that random vector (X,Y) has a joint probability density function
(pdf) given by
︴6(y－x) , if 0 < x < y < 1,
f(x,y) = ︴
︴0 , otherwise
(a) (6 pts.) Verify that f(x,y) is a valid pdf.
(b) (9 pts.) Find the conditional pdf of X|Y = y for any 0 < y < 1.
(c) (10 pts.) Compute Var(X), Var(Y) and Cov(X,Y).
(d) (5 pts.) What is the best constant guess of X under the mean squared
*
error criterion? That is, find the constant a that minimizes
2
E[(X － a) ] over all a∈R.
(e) (5 pts.) What is the best linear prediction of X based on Y under the
_ _
mean squared error criterion? That is, find a + bY that minimizes
2
E[(X － a － bY) ] over a,b∈R.
(f) (5 pts.) What is the best functional prediction of X based on Y under
*
the mean squared error criterion? That is, find g (Y) that minimizes
2
E[(X － g(Y)) ] over functions g on R.
2. (10 pts.) In a class with 15 boys and 10 girls. boys have probability 0.6
of knowing the answer and girls have probability 0.7 of knowing the answer
to a typical question the teacher asks. Assume that whether or not the
students know the answer are independent events. Find the mean and variance
of the number of students who know the answer.
3. The probability mass function of a Poisson(λ) distribution is given by
－λ k
e λ
f(k) = —————, k = 0,1,...
k!
(a) (10 pts.) Show that the moment generation function of a Poisson(λ)
distribution, λ > 0, is given by
t
M(t) = exp[λ(e －1)].
_ 1 n
(b) (10 pts.) Find the probability distribution of X = —Σi=1 X , if X ,
n n i 1
..., X are i.i.d Poisson(λ) random variables.
n
4. (a) (10 pts.) State the Weak Law of Large Numbers for i.i.d samples.
(b) (10 pts.) State the Central Limit Theorem for i.i.d samples.
5. (10 pts.) Suppose that X is a random variable following the Poisson(225)
distribution; i.e.
-225 k
e 225
P(X = k) = ——————, k = 0,1,2,...
k!
Find an approximate value of the probability P(200 ≦ X ≦ 250).