課程名稱︰機率導論
課程性質︰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）︰2013/04/11
考試時限（分鐘）：2:15-3:10pm
試題 :
Introductory Probability
Quiz 2
Thursday 2:15-3:10pm, Aprol 11th, 2013
1. (20 points) The probability of getting a head on a single toss of a coin is
p. Consider that A starts and continues to flip the coin until a tail shows
up, at which poin B starts flipping. Then B continues to flip until a tail
comes up, a which point A takes over, and so on. Let P_n,m denote the
probability that A accumulates a total of n heads befor B accumulates m.
Show that P_n,m = p˙P_n-1,m + (1-p)˙(1-P_m,n).
2. (25 points) A game is played as follows: A random number X is chosen
uniformly from [0,1]. Then a squence Y_1,Y_2,… of random numbers is chosen
independently and uniformly from [0,1]. The game ends the first time that
Y_j>X. If j=1, you lose c dollars. If j≧2, you receive √j dollars.
(a) (12 points) Determine the probability that the game ends at game 3.
Namely, P(X≧Y_1, X≧Y_2, X<Y_3).
(b) (13 points) Find c so that the game is fair theoretically. (Fair means
that the expected winning dollar is zero since you don't pay any fee for
playing this game.)
3. (25 points) The joint probability density function of X and Y is given by
╭ e^-y
│ ── 0<x<y, 0<y<∞
f_X,Y(x,y) = ﹤ y
│
╰ 0 otherwise.
(a) (15 points) Determine f_X|Y(x,y).
(b) (10 points) Compute E[X^3|Y=y].
4. (25 points) Suppos that the life distribution of an item has hazard rate
function
f(t) (t-1)^4
λ(t) = ──── = ──── + 1, t>0.
1 - F(t) 2
What is the probability that
(a) (12 points) the item survives to age 2?
(b) (13 points) a 1 year-old item will survive to age 2?
5. (25 points) We start with a stick of unit length. We break it at a point
which is chosen according to a uniform distribution over [0,1]. For the
piece, of length Y, that contains the left end of the stick, we then repeat
the same process. After the second breaking, let X be the length of the
resulting piece that contains the left end. Now we have three pieces of
stick with lengths X, Y-X, and 1-Y, respectively.
(a) (10 points) Find the joint probability density function of Y and X.
Hint: Note that f_X,Y(x,y) = f_X|Y(x|y) and determine f_X|Y(x|y).
(b) (8 points) Evalute E[X].
(c) (7 points) Find the probability that the three pieces can form a
triangle.