[試題] 100下 林守德 機率 第一次期中考
課程名稱︰機率
課程性質︰必修
課程教師︰林守德
開課學院：電資學院
開課系所︰資訊系
考試日期（年月日）︰2012/4/2
考試時限（分鐘）：3小時
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Probability 2012 Midterm (Prof. Shou-de Lin)
4/2/12 14:30-17:30pm
Total Points: 120
You can answer in either Chinese or English.
1. Please draw the shapes of the following histograms:(a) Symmetric unimodal
(b) Bimodal (c) Positively skewed. (6 pts)
2. X, Y, and Z are three events. Can you propose a real-world example of them
that satisfy both of the following conditions (7 pts):
(a) X and Y are independent.
(b) X and Y become dependent given Z.
3. Given Baye's rule: P(B|A) = P(A|B)*P(B)/P(A). Which term is called the
prior probability? Which term is the Likelihood probability? Which term is
the posterior probability? (6 pts)
4. You are given two red envelops, and told that one contains X dollars and the
other contains 2X. After one of the envelop is picked, you are given a
chance to swap. Assuming the current envelop has Y dollars. if you swap,
there is 1/2 chance you would obtain 2Y and 1/2 chance 0.5Y, therefore the
expectation value of swapping is 1/2(2Y+0.5Y)=1.25Y. if you don't swap, the
expectation value is Y. So you HAVE to swap anyway.
Is this statement correct? If not, can you explan why by using the axioms
of probability(i.e. random experiment, outcome, event, etc)? (8 pts)
5. Suppose there are two companies A and B producing environment detectors.
Company A claims that there is 95% chance their detector alarms when
environment is abnormal. Company B claims that if their detector alarms,
there is 50% chance the environment is abnormal. Suppose the probability of
abnormal environment is 1%. Given the current information, is it possible
to determine which company produces a better alarm? Explain you answer.
(6 pts)
6. You got a binary image, but found that it's too dark to read. You then try
to modify its lightness. Suppose the mean and variance of original image is
50 and 16 respectively in light. You want to change it to 128 and 64
respectively by linear transformation(Y=aX+b). How to determine a and b for
this transformation? (8 pts)
7. if E(X^r)=3^r, r = 1, 2, 3..., find the moment-grnerating function M(t) of
X and the p.m.f. of X. (8 pts)
8. Typist A types 100 words per minute and makes on average 3 errors per
minute; typist B types 150 words per minute and makes on average 4.2 errors
per minute, both following Poisson distribution for making errors. When an
article comes in, we flip a fair coin to determine who should be the typist.
Suppose there is a typed article of 300 words without any error, what is the
likelihood it is typed by A? (8 pts)
9. Let two independent random variables X follow Gamma(α1, θ) and Y follow
Gamma(α2, θ). What is the distribution of X+Y? (5 pts)
10. Given the following random experiments, please comment whether each of
them is likely to produce a random variable that follows a Poisson
distribution, and explain why: (9 pts)
1. Observing the number of people entering CSIE R104 front door from 14:20
-15:00 every Mon.
2. Observing the number of cars passing 長興街警衛亭 every Monday from 10-
11am.
3. Observing the number of cars passing 新生南路忠孝東路交叉口 at 5-6pm
ever Mon.
11. If 10 observations are taken independently from a chi-square distribution
with 19 degrees of freedom, find the probability that exactly 2 of the 10
sample items exceed 30.14. (9 pts)
12. Please describe the relationship between Poisson, Exponential, Gamma, and
Chi-square distributions. (5 pts)
13. in NTU CSIE department, there is a probability course with 3 classes, each
has 50 students. In NxxU CSIE department, they also offer three probability
classes, the chair of NxxU CSIE then claims that their "teaching quality"
is better then NTU CSIE since their class has on the average fewer (33.3
vs 50) students. Do you agree with this statement? Why? (7 pts)
14. (a) There are three boxes: a box containing two gold coins, a box with two
silver coins, and a box with one of each. After choosing a box at random
and withdrawing one coin at random, you find that the chosen coin is a
gold coin. What is the probability that the remaining coin is also gold.
(5 pts)
(b) For the same three boxes: After choosing a box at random, a person
looks at the box and then intentionally reveal one gold coin. Now what is
the probability that the remaining coin is also gold. (5 pts)
15. The mean of a Poisson random variable X is 1. Compute P(1-2σ<X<1+2σ),
where σ is the STD. (6 pts)
16. Show that the geometric distribution is memoryless. (12 pts)
Appendix:
Poisson Distribution
f(x) = (λ^x)*(e^-λ)/x!, μ=λ, σ^2=λ.
Exponential Distribution
f(x) = λ(e^-λx), let θ=1/λ, σ^2=θ^2.
Gamma Distribution
∞
f(x) = (λ^α)*(x^(α-1))*(e^-λx)/Γ(α), where Γ(t) = ∫ y^(t-1)*e^-y dy,
0
let θ=1/λ, M(t)=(1-θt)^-α, μ=αθ, σ^2=α*θ^2.
Chi-square Distribution
f(x) = x^(r/2-1)*e^(-x/2)/Γ(r/2)*x^(r/2), 0≦x＜∞, μ=αθ=(r/2)*2=r,
σ^2=α*θ^2=(r/2)*2^2=2r.
Chi-square table:
(略，查表用)
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