[試題] 99下 張志中 機率導論 第一次期中考
課程名稱︰機率導論
課程性質︰系必修
課程教師︰張志中
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰100/4/13
考試時限（分鐘）：120
是否需發放獎勵金：Y
（如未明確表示，則不予發放）
試題 :
1.(20%) Let S = {1,2,...,n} and 2^S represent the collection of all the subsets
of S. The ecperiment of randomly choosing an element of 2^S is performed in
such a way that each element can be chosen equally likely. This experiment is
carried out twice independently. Denote the outcome of the j-th experiment,
j = 1,2, by X_j 屬於 2^S, i.e. X_j is the (random) subset of S chosen at the
j-th experiment.
(a) Construct a probability model to describe this experiment, which includes
constructing a sample space Ω, a probability measure Ρ, and two
independent random varables X_1 and X_2. You must verify that X_1 and X_2
are indeed independent.
(b) Evaluste Ρ(X_1 屬於 X_2) and Ρ(X_1∩X_2 = ψ).
2.(The Borel-Cantelli lemma)(24%) Given a probability model (Ω,F,Ρ) and a
∞
sequence of events {H_n} , prove the ollowing two statements.
n=1
∞
(a) If ΣΡ(H_n) ﹤∞, then Ρ(limsup H_n) = 0.
n=1 n→∞
∞
(b) If H_1, H_2,... are independent and ΣΡ(H_n) = ∞,
n=1
then Ρ(limsup H_n) = 1.
n→∞
(c) In the probability model of tossing a fair coin indinitely many times,
let H_n be the event that the n-th tossint is a head. Describe the event
limsup H_n verbally.
n→∞
3.(16%) Answer True or False to the following two statements and then explain
or prove your answers.
(a) Let Y be a discrete random variable with probability mass function
p (χ_k) = Ρ(Y = χ_k) = p_k, k = 1,2,3,...
Y
If the infinite sum Σ(χ_k)(p_k) converges, then Ε[Y] exits.
(b) Let Z be a continuous random variable with probability density function f.
M
If lim ∫ zf(z)dz exists, then Ε[Z] exists.
M→∞ -M
4.(30%) 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 gave three pieces of stick with
lengths X, Y-X, and 1-Y, respectively.
(a) Find the joint probability density function Y and X.
(b) Evaluate Ε[X].
(c) fing the probability that the three pieces can form a triangle.
5.(10%) A game is played as follows: A random number X is chosen uniformly
from [0,1]. Then a sequene Y1, Y2,... of random numbers is chosen
independently and uniformly from [0,1]. The game ends the first time that
Yj > X. If j=1, you lose c dollars. If j≧2, you receive √j dollars.
(a) Find c so that the game is fair theoretically.
(b) With th c found above, is it really a fair game? Why or why not?
6.(Bonus problem)(10%) There are two envelope and two numbers, say a and b.
Here a and b, a ﹤b, are tqo fixed real numbers. Each envelope contains one
number. To win the game, one need to choose the envelope containing the
larger correctly. Moreover, one is allowed to choose an envelope at random
and look at the number in it first. Then the opportunity to swop is offered
before making the final decision. Either develop a strategy to win the game
with probability grater than a half, or show that such strategy does not
exist.
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