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）︰103/6/19
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 Markov Chain Property
We are observing and recording the weather in a special place every day. The
possible weather types in the place are sunny, cloudy, and rainy. The
probability of each weather type is dependent of weather type from previous
day which may be modelled as Markov chain. The probability that weather type
in one day is different from the type in previous day is equal to P for each
different type. However, the weather prediction cannot perfectly be modelled
by the assumed Markov model, so the probability that the true weather in one
day is not the predicted weather type is equal to F for each weather type
other than predicted one, which is independent of all previous and future
errors. We use the notion
s_n: Observed sunny day in n-th day.
c_n: Observed cloudy day in n-th day.
r_n: Observed rainy day in n-th day.
S_n: Predicted sunny day in n-th day.
C_n: Predicted cloudy day in n-th day.
R_n: Predicted rainy day in n-th day.
Can the possible weather sequences of our observations be modeled as the
state history of a three-state Markov process?
2 Gaussian Distribution
A noise signal x may be considered to be a Gaussian random variable with an
expected value of 0 and a variance of (σ_x)^2. Assume that any experimantal
value of s will cause an error in a digital communication system if it is
larger than +A.
1. Determine the probability that any particular experimental value of x will
cause an error if
(a) (σ_x)^2 = 10^-2A^2
(b) (σ_x)^2 = A^2
(c) (σ_x)^2 = 9A^2
2. For a given value of A, what is the largest (σ_x)^2 may be to obtain an
error probability for any experimental value of x less than 10^-2? 10^-4?
3 Level of Significance
A random variable x is known to be characterized by either a Gaussian PDF
with E(x) = 4 and σ_x = 0.8 or by a Gaussian PDF with E(x) = 5 and σ_x = 1.
Consider the null hypothesis H_0[E(x)=4,σ_x=0.8]. We wish to test H_0 at the
0.05 level of significance. Our statistic is to be the sum of three
experimental values of random variable x.
1. Determine the conditional probability of false acceptance of H_0.
2. Determine the conditional probability of false rejection of H_0.
3. Determine an upper bound on the probability that we shall arrive at an
incorrect conclusion from this hypothesis test.
4 Probability About Coins
1. For 3600 independent tosses of a fair coin, determine the probability that
the number of heads is within ±1% of its expected value.
2. The outcomes of successive flips of another particular coin are dependent
and are found to be described fully by
Prob(H_n+1|H_n) = 3/4 Prob(T_n+1|T_n) = 2/3
where we have used the notation
Event H_k: Heads on k-th toss Event T_k: Tails on k-th toss
We know that the first toss came up heads.
(a) Determine the probability that the first tail will occur on the k-th
toss (k=2,3,4,...)
(b) What is the probability that flip 5000 will come up heads?
(e) Given that flips 5001, 5002, ..., 5000+m all have the same result, what
is the probability that all of these m outcome are heads? Simplify your
answer as much as possible and interpret your result for large values
of m.
3. Consider the problem of estimating the parameter P (the probability of
heads) for one more particular coin. To begin, we assume the following
a priori probability mass function for P:
0.2 P_0 = 0.4
p_P(P_0) = { 0.6 P_0 = 0.5
0.2 P_0 = 0.6
We are now told that the coin was flipped n times. The first flip resulted
in head, and the remaining n-1 flips resulted in tails.
Determine the a posteriori PMF for P as a function of n for n >= 2. Prepare
neat sketches of this function for n = 2 and n = 5 and briefly state your
findings.
註：考卷有附 normal distribution table