[試題] 106-2 機率 林守德
課程名稱︰機率
課程性質︰資訊系必修(單班)
課程教師︰林守德
開課學院：電機資訊學院
開課系所︰資訊工程學系
考試日期（年月日）︰2018/4/26 14:20~17:20 pm
考試時限（分鐘）：180
試題 :
Probability2018 Midterm (Prof. Shou-de Lin)
4/26/2018 14:20-17:20pm
Total Points: 120
You can answer in either Chinese or English
1. [Short Answers 15pts]
(a) What are Prior and Posterior probabilities?
(b) What is Simpson's Paradox?
(c) What's the relationship between normal distribution and
Gamma distribution?
2. [CDF 7pts]
Below are the numbers drew independently from certain continuous
distribution. Can you plot the CDF?
[0.2 , 0.5 , 0.5 , 0.2 , 0.3 , 0.4 , 0.5 , 0.1]
3. [Axioms of Probability 9pts]
We know that probability (of an event) has to be defined given a
random experiment that is repeatable. However, we often hear sports
fans say something like "There is a 70% chance that X will win the
battle tomorrow". Since TOMORROW is non-repeatable, can you explain
what they really mean by saying the chance is 70%?
4. [Conditional Probability 7pts]
A multiple choice has 4 choices for each question. A student John has
50% chance to know the answer to the question, 25% he doesn't exactly
know the answer but can eliminate one choice, and 25% he has no idea
what could be the right choice. The teacher know that John answer one
question correctly, what's the probability he really knows the answer?
5. [Poisson 7pts]
A hotel has two phone lines for reservation. The number of calls coming
per minute into each line is i.i.d. and follows a Poisson distribution
with mean 3. What is the probability that at least two reservation
requests arrive in a minute?
6. [Expectation 7pts]
What is the probability that a random vatiable X is less than its
expected value, given X has an exponential distribution with parameter θ?
7. [Continuous RV 10pts]
Suppose that X is a continuous random variable with
2
2018t+512t
Ｍ_X(t) = e
X
(a) What is the probability density function of Y = e ?
(B) What is E[Y] ?
8. [Multivarite 12pts]
Suppose the X , X , ... , X are discrete random variable,
1 2 r
where r > 2 and have a joint probability mass fuction f :
n! x1 x2 xr
f(x, x , ...,x ) = --------------- P P ... P ,
1 2 n x !x ! ...x ! 1 2 r
1 2 r
where x , x , ... , x ∈{0,1,2,...} , x + x + ... + x = n > 0
1 2 r 1 2 r
p , p , ... , p > 0 , p + p + ... + p = 1.
1 2 r 1 2 r
(a) What is the marginal probability mass function of X ?
1
(b) Prove or disprove that X and X are indepentdent.
1 2
10. [Recursive 10pts]
A teacher T decides to give a prize among 11 students by the
following game. The teacher and 11 students sit cyclically, and
the person who currently takes the prize should pass it to either
the left-hand-side or the right-hand-side person with equal pobability.
( P(give to lfs) = P(give to rfs) = 1/2 ) Repeat the passing rule until
all of the students have touched the prize, and the last student touching
it will win the prize.
If the passing procedure start from the teacher, which seat should you
choose to be the winner? Please use probability to justify your answer.
Ｔ
11 １
10 ２
９ ３
８ ４
７ ５
６
11. [Continuius RV 10pts]
The city transportation authority is claiming that they schedule 4 buses
per hour on the average(according to certain unknown distrubution)
i.e. about 1 bus every 15 minutes on average. However, major Ko doen't
believe it, since he thinks he usually waits for longer amount of time.
So, he asked many people at the bus stop how long they have been waiting
till the next bus arrives. He found out that the average waiting time is
larger than 15 mins. Assuming the people are coming randomly according to
uniform distribution, explain why the average waiting fime for the
passenger is larger than the average duration of a bus.
12. [Bayes Theorem 12pts]
Company 1 announces a disease (occur rate = 20%) testing product T1.
Company 2 also announces a testing product T2 for the same disease.
The performance of them look like:
P(T1=positive | Disease=true) = 0.7 , P(T1=negative | Disease=false) = 0.7
P(T2=positive | Disease=true) = 0.9 , P(T2=negative | Disease=false) = 0.6
Q1: A careless doctor performed a test on a patient and found that the
result is positive. However, this doctor forgot which testing product
was chosen. Can you tell this doctor which product is more likely to
be the one used given positive result?
Q2: If a patient has been tested positive on both products, what is the
probability that he/she really has the disease? (assuming that the
test results are conditionally independent givedisease)
---------------------------------------------------------------------------
Poisson Distribution :
x -λ
λ e
f(x) = ----------
x!
Exprnential Distribution :
-λx
f(x) = λe , let θ = 1/λ
MGF for normal distribution :
2 2
μt+σt /2
e
--
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※ 編輯: bluedog666 (140.112.211.204), 04/27/2018 23:56:16
推
04/28 00:13,
6年前
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04/28 00:13, 1F