課程名稱︰機率導論
課程性質︰數學系必修
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）︰2017/5/11
考試時限（分鐘）：30分鐘
試題 :
Quiz 4 (2017/5/11)
1.(15%) Suppose that 3 balls are chosen without replacement from an urn
consisting 4 white and 6 red balls. Let X equal 1 if the i-th ball selected
i
is white, and let it equal 0 otherwise. Give the joint probability mass
function of X , X , X .
1 2 3
2. If X and Y are independent exponential random variables with respective
parameters λ , λ .
1 2
(a)(15%) Find the probability density function of Z=X/Y.
(b)(10%) Find P{ X ＜ Y }.
3. Suppose that random vector (X,Y) has a joint probability density function
(pdf) given by
－y
╭ C(y－x)e , if －y＜x＜y, 0＜y＜∞,
f(x,y) = │
╰ 0 , otherwise
for some constant C.
(a)(10%) Find the value of the constant C.
(b)(15%) Find the marginal probability density functions of X and Y.
(c)(15%) Find E[X] and E[Y].
4.(20%) Suppose that random vector (X,Y) has a joint pdf given by
╭ 24xy , if 0＜x＜1, 0＜y＜1, 0＜x＋y＜1,
f(x,y) = │
╰ 0 , otherwise.
Are X and Y independent?