BÀI TẬP LỚN MÔN XSTK NHÓM 10-2023
Chú ý: h là số thứ tự trong bảng danh sách điểm danh. SV phải ghi số thứ tự h sau tên họ của mình.
Phải thay h bằng số thứ tự trước khi tính toán.
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Problem 1: An experiment consists of tossing two dice.
(a) Find the sample space .
(b) Find the event A that the sum of the dots on the dice equals 7.
(c) Find the event B that the sum of the dots on the dice is greater than 10.
(d) Find the event C that the sum of the dots on the dice is greater than 12.
Problem 2: Two manufacturing plants produce similar parts. Plant 1 produces 1,000 parts, 100 of which
are defective. Plant 2 produces 2,000 parts, 150 of which are defective. A part is selected at random
and found to be defective.What is the probability that it came from plant 1?
Problem 3: There are n persons in a room.
(a) What is the probability that at least two persons have the same birthday?
(b) Calculate this probability for n = 50.
(c) How large need n be for this probability to be greater than 0.5?
Problem 4: Romeo and Juliet have a date at a given time, and each will arrive at the meeting place with a
delay between 0 and 1 hour, with all pairs of delays being equally likely. The first to arrive will wait for
15 minutes and will leave if the other has not yet arrived. What is the probability that they will meet?
Problem 5: Radar detection. If an aircraft is present in a certain area, a radar correctly registers its
presence with probability 0.99. If it is not present, the radar falsely registers an aircraft presence with
probability 0.10. We assume that an aircraft is present with probability 0.05.
a. What is the probability of false alarm (a false indication of aircraft presence), and the probability
of missed detection (nothing registers, even though an aircraft is present)?
b. Let A ={an aircraft is present}, B ={the radar registers an aircraft presence}. We are given that
P(A) = 0.05, P(B |A) = 0.99, P(B |Ac) = 0.1. What is the probability of aircraft present with condition
radar registers?
Problem 6: You enter a chess tournament where your probability of winning a game is 0.3 against half
the players (call them type 1), 0.4 against a quarter of the players (call them type 2), and 0.5 against the
remaining quarter of the players (call them type 3). You play a game against a randomly chosen
opponent. What is the probability of winning?
Problem 7 The annual snowfall at a particular geographic location is modeled as a normal random
variable with a mean of μ = 60 inches, and a standard deviation of σ = 20. What is the probability that
this year’s snowfall will be at least 80 inches?
(x a )2
2
Problem 8: The Rayleigh density of random variable X is fX (x )
b
b (x a )e if x a
0 if x a
a. Find the distribution function FX (x ) .
b. The lifetime of a system expressed in weeks is a Rayleigh random variable X for which
a 0,b 400 . What is the probability that the system will not last a full week? What is the probability
that the system will exceed one year?
Problem 9: (The Quiz Problem). Consider a quiz game where a person is given two questions and must
decide which question to answer first. Question 1 will be answered correctly with probability 0.8, and the
person will then receive as prize $100, while question 2 will be answered correctly with probability 0.5,
and the person will then receive as prize $200. If the first question attempted is answered incorrectly, the
quiz terminates, i.e., the person is not allowed to attempt the second question. If the first question is
answered correctly, the person is allowed to attempt the second question. Which question should be
answered first to maximize the expected value of the total prize money received? (This example, when
generalized appropriately, is a prototypical model for optimal scheduling of a collection of tasks that have
uncertain outcomes.)
Problem 10: If the weather is good (which happens with probability 0.6), Alice walks the 2 miles to class
at a speed of V =5 miles per hour, and otherwise drives her motorcycle at a speed of V =30miles per
hour. What is the mean of the time T to get to class?
Problem 11: Discrete random variable X , Y have a joint distribution table
Y
0,2 0,4 0,6 h
X
1 0,01+0,001. h 0,02 0,08-0,001. h 0,04
2 0,03 0,20 0,15 0,06
3 0,09-0,001. h 0,05 0,08+0,001. h 0,08
4 0,02 0,04 0,03 0,02
a. Find the both marginal distrbution table of X and Y .
b. Are X and Y statistically independent?
c. Find the probability mass function of X with the condition Y h .
d. Find E X , EY ; E X Y h ; D X , DY and XY .
Problem 12: Consider a transmitter that is sending messages over a computer network. Let us define the
following two random variables:
X : the travel time of a given message, Y : the length of the given message.
We know the PMF of the travel time of a message that has a given length, and we know the PMF of the
message length. We want to find the (unconditional) PMF of the travel time of a message.
We assume that the length of a message can take two possible values: y 102 bytes with probability 5/6,
5 / 6 if y 102
4
and y 10 bytes with probability 1/6, so that pY (y )
1 / 6 if y 104
We assume that the travel time X of the message depends on its length Y and the congestion level of the
network at the time of transmission. In particular, the travel time is 104Y secs with probability 1/2,
103Y secs with probability 1/3, and 102Y secs with probability 1/6. Thus, we have
1 / 2 if x 102 ,
1 / 2 if x 1,
pX |Y (x | 102 ) 1
1 / 3 if x 10 ,
4
pX |Y (x | 10 ) 1 / 3 if x 10,
1 / 6 if x 1,
1 / 6 if x 100,
Determine the PMF of X .
Problem 13: Consider an experiment of drawing randomly three balls from an urn containing two red,
three white, and four blue balls. Let ( X ,Y ) be a bivariate r.v. where X and Y denote, respectively, the
number of red and white balls chosen.
a. Find the range of ( X ,Y ).
b. Find the joint pmf's of ( X ,Y ).
c. Find the marginaI pmf's of X and Y .
d. Are X and Y independent?
Problem 14: The joint pmf of a bivariate r.v. ( X ,Y ) is given by
2
xi 1,2; y j 1,2, 3
cx y
pXY (xi , y j ) i j
0 otherwise
where c is a constant.
a. Find the value of c .
b. Find the marginal pmf's of X and Y .
c. Are X and Y independent?
Problem 15:
John throws a dart at a circular target of radius r. We
assume that he always hits the target, and that all
points of impact (x, y) are equally likely, so that the
joint PDF of the random variables X and Y is
uniform. What is the marginal PDF fY (y ) and the
conditional PDF fX |Y (x | y ) ?
Problem 16: Let the random variables X and Y have a joint PDF
2 if x 0; y 0; x y 1
fXY (x , y )
0 otherwise
What is the conditional density of X given Y = y and E[X | Y = y]?
Problem 17:
e. Determine the value of c that makes the function f (x , y ) c(x y ) a joint probability density
function over the range 0 x 3 and x y x 2 of random vector (X ,Y ) .
f. Determine the following:
P X 1,Y 2 P 1 X 2 P Y 1 P X 2,Y 2 E X
g. Determine the following:
(i) Marginal probability distribution of X
(ii) Conditional probability distribution of Y given that X 1
(iii) E Y | X 1
(iv) P Y 2 | X 1
(v) Conditional probability distribution of X given that Y 2 .
Problem 18: Let (X1,..., X n ) be a random sample of a Poisson r.v. X with unknown parameter .
1 n 1
a. Show that n Xi and 2 X1 X2 are both unbiased estimators of .
n i 1 2
b. Which estimator is more efficient?
Problem 19: The diameters of certain cylindrical items produced by a machine are r.v.’s distributed as
N(μ, 0.01). A sample of size 16 is taken and is found that x = 2.48 inches. If the desired value for μ is 2.5
inches, formulate the appropriate testing hypothesis problem and carry out the test if α = 0.05.
Problem 20: In a certain university 400 students were chosen at random and it was found that 95 of them
were women. On the basis of this, test the hypothesis H that the proportion of women is 25% against the
alternative A that is less than 25% at level of significance α = 0.05.
