課程名稱︰機率
課程性質︰必修
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）︰2014/4/14
考試時限（分鐘）：180
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Total Point : 120
You can answer in either Chiese or English
1.[Independency]
X,Y,and Z are three random variables. Can you proposal a real-world example
of them that satisfy the following conditions:
(a) X and Y are independent, but becomes dependent given Z (5pts)
(b) X and Y are dependent, but becomes independent given Z (5pts)
2.[Application on Probability]
There is a group of n students who occupied the Legislative Yuan (LY). The
observation is that: at each minute, there is 1/2 chance one student will
leave LY (i.e. n=n-1), and 1/2 chance one student enters LY (i.e. n=n+1).
The police make a deal with the leader: If they reach 2n students before
everybody leaves LY (n=0), then the leader will not be arrested. However, if
the number reaches 0 before it becomes 2n, then the leader will be arrested
immediately.
(a) Given t minutes passed, what is the chance that the leader has
already been arrested ? (12pts, if you cannot derive the close
form solution, please describe how to generate this probability)
(b) If t >> n, what is the answer for this question ? (5pts)
3.[Probability and Conditional Probability]
(a) In a modified Monty Hall Problem, assuming there are n doors and behind
n-k of them are goats, while the remaining k (k << n) is a car. After a
participant picks a door, the host (who knows where the cars are) will
intentionally open a door with goat. In this case, should the
participant swap his current choice with one of the remaining door?(5pts)
(b) If the host does not know where are the car, and he opens a door with a
goat. Should the participant swap ? (5pts)
Please explain your answers using probability.
4.[Probability and expectation]
A grocery store has available n watermeleons to sell and makes $1.00 on each
sale.Say the number of consumers of these watermelons is a random variable
that has a distribution that can be approximated by
f(x) = 1/200 , 0 < x < 200,
a p.d.f. of the continuous type. If the grocer does not have enough
watermelons to sell to all consumers, she figure that she loses $5.00 for
each unhappy customer. But if she surplus watermelons, she loses 50 cents on
each extra watermelon. What should n be to maximize "profit"? (10pts)
5.[Random Experiment]
You are asked to design a random experiment to estimate the circumference
ratio π. The only function you can use is the random-value-generator
random(), which returns a value between [0,1]. Please describe your
experiment (you can use pseudo code or simply explain it in plain text).
(10pts, Note that you can use while, if, and +-*/ in the pseudo code)
6.[Bayes rule]
Company1 announces a disease (occur rate = 20%) testing product T1. The
performance looks like:
P(T1 = positive | Diease = true ) = 0.7
P(T1 = negative | Diease = false) = 0.7
Company2 also announces a testing product T2 for the same disease. The
performance looks like:
P(T2 = positive | Diease = true ) = 0.9
P(T2 = negative | Diease = false) = 0.6
Q1: A careless doctor performes a test on a patient and found that the result
is positive. However,this doctor forget which testing product was chosen.
Can you tell this doctor which product is more likely to be the one used
given positive result ? (5pts)
Q2: If a patient has been tested positive on both products, what is the
probability that he/she really has the diease (assuming that the test
results are conditionally independent given diease) ? (5pts)
7.[Exponential Distribution]
Let X be a random variable that represents the number of days that it takes
a high-risk driver to have an accident. Assume that X has an exponential
distribution. If P(X<50)=0.25, compute P(X>100 | X>50). (5pts)
8.[Poisson]
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 :
(a) Observing the number of people entering CSIE R104 front door from
14:20-15:00 every Monday
(b) Observing the number of cars passing 長興街警衛亭 every Monday
from 10-11am
(c) Observing the number of cars passing 新生南路忠孝東路交叉口 at
5-6pm every Monday
(9pts)
9. Amy wants to buy the May-Day concern ticket online. Since it is really hard
to get one, she used five computers (C1~C5) to in pareallel to ensure she
can get it. But there are different successful rate for each computer to get
the ticket. The successful rate of Ci computer is P(i+1) = 1.05*P(i)-0.05. A
'round' is defined as one trail for every computer. A random variable X is
defined as the number of rounds required to get one ticket. We know the
variance of X is 2. So what's P(1) ? (7pts)
10. In the movie theater there are 300 seats. 300 people including Jack lined
up to enter the theater. Jack is the 200th person to enter the theater, but
when it's his term, he found that his seat number faded so he has no idea
where to seat. Jack decides to randomly pick the seat because everyone was
so nice that if their seat is taken, they will simly find other seat
randomly. What's the probability of the last person to seat his/her own seat
correctly ? (10pts)
11. One day Takasi found that around him there were four zombies. He picked up
the gun to kill them. Since it becomes easier to kill a zombie if there are
more around, the probability for Takasi to kill the zombie if n/5 (n is the
number of zombie) and he can only kill one zombie in a time. When the zombie
was killed, the zombie will have 30% chance to revived by their friend. That
says, if there is only one zombie left, it won't rebirth. What's the
expected time Takasi needed to kill all the zombies ? (10pts)
12. In the MMORPG world, Shiroe and his partners wants to meet with the king.
They started at "A" and is allowed to choose only three directions (right,
up, down), with probability becomes P(right) = 1/2, P(up) = P(down) = 1/4.
However, if he can't go up or down, the probability becomes P(right) = 1/2,
P(up or down) = 1/2. In the graph there are many traps in each area, denoted
as 'X'. Entering 'X' means game over. One special place is the rest area,
there they can choose to start at "B", "C" and "D". Assuming Shiroe took
SD's probability couse previously and can make the best decision, what's the
probability that Shiroe meets the king safely ? (12pts)
┌─┬─┬─┬───┬─┬─┬─┬───┐
│O │O │O │ │B │O │O │ │
├─┼─┼─┤ ├─┼─┼─┤ │
│A │O │X │ Rest │C │O │X │ King │
├─┼─┼─┤ ├─┼─┼─┤ │
│O │O │O │ │D │O │O │ │
└─┴─┴─┴───┴─┴─┴─┴───┘
------------------------------------------------------------------------------
Poisson distribution
λ^x e^-λ
f(x) = ─────
x !
μ = λ, σ^2 = λ
(1-p)
The variance of geometry distribution is f(x) = ────
p^2
Exponential Distribution
f(x) = λe^(-λx), let θ = 1/λ, μ = θ , σ^2 = θ^2
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