課程名稱︰機率
課程性質︰必帶
課程教師︰劉長遠
開課學院：電資學院
開課系所︰資訊系
考試日期（年月日）︰103/4/17
考試時限（分鐘）：14:20 ~ 17:20
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1 Probability basics
1. Random variables A, B, C, and D satisfy following conditions:
(a) P(A)<1, P(B)<1, P(C)<1, P(D)<1.
(b) A and B are independent, and C and D are independent.
(c) A and C are independent.
Are B and D independent? Why?
2. Random variables A, B, C, and D satisfy following conditions:
(a) P(A)<1, P(B)<1, P(C)<1, P(D)<1.
(b) A and B are mutual exclusive, and C and D are mutual exclusive.
(c) A and C are independent.
Are B and D independent? Why?
3. In communication network, link failures are independent, and each link has
a probability of failure of P. Consider the physical situation before you
write anything. A can communicate with B as long as they connected by at
least one path which contains only in-service link.
a b c d
┌─┤├─┬─┤├─┬─┤├─┬─┤├─┐
│ │ │ │ │
│ │ │ │ │
A ──┤ ┴ ┴ ┴ ├── B
│ ┬ i ┬ j ┬ k │
│ │ │ │ │
└─┤├─┴─┤├─┴─┤├─┴─┤├─┘
e f g h
Figure 1: The communication network between A and B.
(a) Given that exactly 7 links have failed, determine the probability that
A can still communication with B.
(b) Given that a, h, and j have been failed without information about
conditions of other links, determine the probability that A can still
communicate with B.
2 Random variables
1. We are given the following information about random variable x:
E[x] = 24, E[x^2] = 625,
E[x|x > E[x]+σ_x], Prob[x > E[x]+σ_x] = 0.2.
Determine the numerical value of E[x|x <= E[x]+σ_x].
2. Discrete random variable x is described by the PMF
K - x_0/15 if x_0 = 0,1,2
p_x(x_0) = ｛
0 for all the other values of x_0
Let d_1, d_2, ..., d_N represent N successive independent experimental
values of random variable x.
(a) Determine the numerical value of K.
(b) Determine the probability that d_1 > d_2.
N
(c) Determine the probability that Σ d_i<=1.0.
i=1
3 Transform
Let k be a random variable with z-transform:
T
P_k(z) = 0.25z + 0.25z^4 + 0.5e^-2(z-1)(z-2)
T
Let x be a random variable with s-transform: f_x(s) = 0.5/(s + 0.5)
Let w be the sum of k independent experimental values of random variable x.
1. E(k^2)
2. E(w)
4 Poisson Process
A woman is seated beside a conveyer belt, and her job is to remove certain
items from the belt. She has a narrow line of vision and can get these items
only when they are right in front of her.
She has noted that the probability that exactly k of her items will arrive
in a minutes is given by
p_k(k_0) = 2^k_0e^-2 / k_0! k_0 = 0,1,2,3,...
and she assumes that the arrivals of her items constitute a Poisson process.
1. If she wishes to sneak out to have a beer but will not allow the expected
value of the number of items she misses to be greater than 5, how much
time can she take?
2. If she leaves for two minutes, what is the probability that she will miss
a total of exactly 4 items?
3. The union has installed a bell which rings once a minute with precisely
one-minute intervals between gongs. If, between two successive gongs, more
than 3 items come along the belt, she will handle only 3 of them properly
and will destroy the rest. Under this system, what is the probability that
any particular item will be destroyed?