课程名称︰机率
课程性质︰必修
课程教师︰林守德
开课学院：电资
开课系所︰资工
考试日期（年月日）︰2015/06/25
考试时限（分钟）：180
试题 :
Probability2015 Final (Prof. Shou-de Lin)
6/25/2015 14:30-17:30pm
Total Points: 120
You can answer in either Chinese or English.
◆ Note, please use Φ function as the CDF of standard distribution (no need
to calculate the correct value). For instance, P(X<1) given standard normal
distribution can be represented using Φ(1). Also Φ(2)=98%, Φ(1.65)=95%,
◆ Variance of Binomial distribution: np(1-p)
◆ KL divergence:
D_KL(P||Q) = Σ_i ln(P(i)/Q(i))P(i).
1. John builds an auto-translation tool, and wants to claim that the tool he
builds is better than the Google Translator (whose accuracy is 75%). He
tested on 100 sentences and found 76 of them were translated correctly.
(a) What is the 95% confidence interval of the accuracy of his system
(hint: Z_0.05=1.65, Z_0.025=1.96)? Can you use confidence interval to
convince people John's system is better? (6pts)
(b) Given H_0=75%, That is the p-value? You don't need to calculate the
exact p-value, just list its equation is good enough. (6pts)
2. Suppose that A1 and A2 are independent uniform random variables on [0, 1].
Let X = max{A1, A2}. Compute the probability density function for X. (6pts)
3. X follows a uniform distrubution on [0, 6].
Y follows exponential distribution with Θ = 1/3; Also X, Y are independent.
Find the probability such that Y>=X. (6pts)
4. Please describe how to estimate the value of π (i.e 3.1415...) 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: ) (10pts)
5. Calculate the Pagerank of A, B, C, D or provide a pseudo code telling us how
to find the Pagerank values of A, B, C, D (8pts)
B
↙↑ ↘
A ←┼─ C
↘↖│ ↗
D
6. You want to build an automatic process to correct "Chiglish sentences"
(中式英文) with misused verb. For example, you want the system to
automatically correct the sentence "open the light" to "I turn on the
light". You are given a large set of English sentences written by native-
Chinese writers, and a large set of sentences from native English speakers.
Please describe how you can use the n-gram language model together with
noisy channel model to resolve this problem. (8pts)
7. Consider throwing 60 four-sided unbiased dices all together at a time and
observe their total sum. Assume that you perform the experiment 2000 times.
After the experiments, you plot the observed counts corresponding to all
possible sums (from 60 to 240). Give the continuous function that you think
best fits the observe data plot (8pts)
8. Let X have a gamma distribution with α=2 and θ=2. Please find the
probability density function of Y=e^2X (8pts)
9. Let X_1, X_2, X_3...X_n be iid random sample from geometric distribution
with unknown parameter p, please use maximum likelihood function to find
this p (8pts)
10. (a) Y=x1+x2, and Mgf(x1)=(e^t -1)/t Mgf(x2)=t(e^(t^2/2)/(e^t -1)), use
Chebyshef's inequality to find the lower bound of P(-2<Y<2) (7pts)
(b) Do you think this lower bound attained from Chebyshef's inequality is
tight? Explain your answer (hint: use the lookup table we provide) (7pts)
11. Suppose two random points are chosen independently and uniformly from the
unit square [0, 1] * [0, 1], and let D denotes the distance between the two
points (X_1, Y_1)(X_2, Y_2).
D^2 = (X_1-X_2)^2 + (Y_1-Y_2)^2
Find E(D^2). (8pts)
12. The table shows the choices made by 120 players of a Rock-Paper-Scissors
game. A player gains an advantage in playing this game if there is evidence
that the choices made are not equally distributed among the three options.
Use a chi-square test to see if there is evidence that any of the
proportions are different from 1/3. (8pts)
┌────────┬──────┐
│Option selected │Frequency │
├────────┼──────┤
│Rock │60 │
├────────┼──────┤
│Paper │39 │
├────────┼──────┤
│Scissors │21 │
└────────┴──────┘
13. Given a 6-side dice, what is the mutual information between the top side
and the front face (the side most facing you)? (8pts)
14. (Entropy) (8pts)
Let X be a function of Y. Prove that H(X) <= H(Y).