課程名稱︰機率
課程性質︰資訊系必修
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）︰2015/04/23
考試時限（分鐘）：180
試題 :
編按: 題目有部分修正，將直接寫到題目之中，修改部分將以紅字表示
Total Points: 120
You can answer in either Chinese or English
1. [MGF] 7pts
Find P(|X|≦1), given that X has moment-generating function:
M(t) = (111/345)e^-2t + (1/3)e^-1t + (1/5)e^t + (41/203)e^2t
+ (1201/27807)e^3t
2. [Probability and Conditional Probability] 10pts
(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) are cars. 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?
Please explain your answer using probability.
(b) If the host does not know where the cars are, and he opens a door with a
goat. Should the participant swap? Please explain your answer using
probability.
3. [Random Experiment] 10pts
The city transportation authority is claiming that they schedule 4 buses per
hour on the average (according to certain unknown distribution) i.e. about 1
bus every 15 minutes on average. However, major Ko doesn'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 min.
Explain why the average waiting time for the passenger is larger than the
average duration of a bus.
4. [Events] 7pts
Let E, F, and G are mutually exclusive events of a random experiment (RE)
with probability P(E), P(F), P(G). Suppose that a new random experiment is
designed as repeated the original RE until event E, F, and G occur. What is
the probability that event E occurs before event F?
5. [permutation] 10pts
S1, S2, ..., Sn are playing a game. In the beginning they line up from S1
(first) to Sn (last). Then they in turn (start from S1) throw a 6 side fair
dice. When the dice obtain 1 or 2, the person go to the beginning of the
line, when the dice obtain 3 or 4, the person stay the current position in
the line, when the dice obtain 5 or 6, the person go to the end of the line.
When a person go to the beginning or end of the line, the person behind him
fill the left position.
After all person throw the dice, what is the probability that the order of
the line is not changed (the same as the initial line)?
6. [Poisson] 10 pts
Please describe how to estimate the value of e (i.e. 2.71828...) using only a
random function r() that returns a real number between [0, 1], and operation
+, -, *, /. Please write a C or pseudo code to do so (hint: 'e' appears in
the Poisson distribution).
7. [Multivariate] 10 pts
Suppose that A1 and A2 are independent uniform random variables on [0, 1].
Let X = max{A1, A2} and Y = min{A1, A2}. Compute the following:
(a) The probability density function for X and Y.
(b) The expectation E[X] and E[Y].
8. [continuous RV] 10 pts
X follows a uniform distribution on [0, 6].
Y follows exponential distribution with θ = 1/3; Also X, Y are independent.
(a) Find the mean and variance of X+5Y.
(b) Find the probability such that Y ≧ X.
9. [Bayes Rule] 10 pts
Jennifer checks the weather forecast before deciding whether to carry an
umbrella most of the mornings. If the forecast says rain, then the
probability of actually having "rain" that day is 70%. On the other hand, if
the forecast says "no rain", then the probability of actually having rain is
10%. During spring/summer the forecast is "rain" 15% of the time and during
fall/winter it is 80%. Jennifer has a probability of 40% to miss the
forecast. When missing the forecast, she has a 50% chance to bring an
umbrella. Otherwise she will bring an umbrella whenever the forecast says
"rain".
(a) One day, Jennifer missed the forecast and it rained. What is the
probability that the forecast was "rain" given it was during the spring?
(b) Jennifer is carrying an umbrella and it's not raining. What is the
probability that she saw the forecast? Does it depend on the season?
10. [correlation coefficient] 10 pts
In this question, we want to show that the correlation coefficient is a
measure of the amount of linearity.
X and Y are discrete random variables that take values from (-∞, +∞) and
their outcome space is of the same size. They can form pairs from (X_0, Y_0),
(X_1, Y_1), to (X_n, Y_n). Let us now consider all possible lines in 2D space
passing through the point (μ_x, μ_y), that is the mean of X and Y. Such
line can be expressed as the form y-μ_y = b(x-μ_x), where b is the slope.
Thus, the vertical distance from any point (X, Y) to this line can be denoted
as |Y-μ_y-b(X-μ_x)|.
(1) Find the b = b_m that minimizes E[(|Y-μ_y-b(X-μ_x)|)^2], which is the
expectation of the square of vertical distance to the line. Please
express b_m using the correlation coefficient ρ and σ_x, σ_y.
(2) Let E[(|Y-μ_y-b(X-μ_x)|)^2] = K(b), what is K(b_m)? Use it to show that
ρ lies in [-1, 1] and whyρ is a measure of linearity.
11. [discrete r.v] 10 pts
B02 consists of X boys and Y girls, they have different taste of movies. Let
X_1, X_2, X_3 be the number of boys who want to see Comic (X_1), Action
(X_2), and both of them (X_3), Y_1, Y_2, Y_3 for girls respectively, where
X = X_1 + X_2 + X_3, Y = Y_1 + Y_2 + Y_3. We hold a blind date split B02 into
pairs. Everybody has the same chance to group with others and there is no one
left. Note that the conditional probability of a successful pair is defined
in the below table. It says that if the pair is of same sex and wants to see
the same movie, there is 50% chance it is a successful match, and so on.
┌───────┬────────┬────────┐
│ │Same taste │Different taste │
├───────┼────────┼────────┤
│Same sex │0.5 │0.1 │
├───────┼────────┼────────┤
│Different sex │1 │0.7 │
└───────┴────────┴────────┘
What is the expected number of successfully matched pairs (Hint: The method
used in problem 2.7 will do you a favor)?
12. [conditional pmf] 16pts
Suppose our 3 TAs have built a website containing lots of tools for preparing
midterm. The TAs also need to correct data from the website (each TA have 0.5
chance to correct the data from website). Let random variable Y be the number
of TAs who are correcting data on the website. However, the website's
bandwidth can only allow at most α-2Y students downloading, where α is an
integer larger than 6. The website crashes when there are more than α-2Y
students downloading. We have known that 90% of the students are coming for
surfing while 10% are for downloading (Once students in, they do not leave,
so do TAs). Let X denotes the total number of students the website serves
before shutting down,
(1) What is P(Y)? (2 pts)
(2) What is P(X|Y)? (6 pts)
(3) What is the expected number of X given that there are more than one TA
correcting data on the website? (8 pts)
===============================================================================
Cov (X_1, X_2) = E[(X_1 - μ_1)(X_2 - μ_2)],
ρ= Cov (X_1, X_2) / (σ_1σ_2)
Poisson Distribution
f(x) = (λ^x e^-λ)/x!, μ=λ, σ^2=λ
Exponential Distribution
f(x) = λe^-λx, let θ=1/λ, μ=θ, σ^2 = θ^2