## Description

Exercise 1.

Calculate the infinite series

X∞

k=0

k ·

2

3

k+1

c 2020 Stanley Chan. All Rights Reserved. 1

Exercise 2.

Evaluate the integrals

(a)

Z b

a

1

b − a

x −

a + b

2

2

dx

(b)

Z ∞

0

λxe−λxdx

(c)

Z log y

− log y

λx

4

e

−λ|x|

dx,

where y > 1.

c 2020 Stanley Chan. All Rights Reserved. 2

Exercise 3.

Evaluate the infinite series

X∞

k=0

(k − λ)

2 λ

k

e

−λ

k!

(1)

c 2020 Stanley Chan. All Rights Reserved. 3

Exercise 4.

Simplify the following sets with the domain of real numbers in mind

(a) [2, 5] ∩ ([1, 3] ∪ {0, 3, 4})

(b) (1, 2)c ∪ [4, 6]

(c) T∞

n=1(2 − 1/n, 2 + 1/n)

(d) S∞

n=1[3, 6 −

1

n

]

c 2020 Stanley Chan. All Rights Reserved. 4

Exercise 5.

A space S and three of its subsets are given by S = {1, 3, 5, 7, 9, 11}, A = {1, 3, 5}, B = {7, 9, 11}, and

C = {1, 3, 9, 11}. Find (a) A ∩ B ∩ C, (b) Ac ∩ B, (c) A\C, and (d) (A\B) ∪ B.

Exercise 6.

Prove the second part of DeMorgan’s Law, i.e., show that (A ∪ B)

c = Ac ∩ Bc

.

c 2020 Stanley Chan. All Rights Reserved. 5

Exercise 7.

Let A = (−∞, r] and B = (−∞, s] where r ≤ s. (a) Find an expression for C = (r, s] in terms of A and B.

(b) Show that B = A ∪ C, and A ∩ C = ∅.

Exercise 8.

Show that if A ∪ B = A and A ∩ B = A, then A = B.

c 2020 Stanley Chan. All Rights Reserved. 6

Exercise 9.

This is a programming exercise. You can use either MATLAB or Python.

(a) Compute the result of the following matrix vector multiplication.

1 2 3

4 5 6

7 8 9

×

1

2

3

(b) Plot a sine function on the interval [−π, π] with 1000 data points using matplotlib.pyplot.plot in

Python or plot in MATLAB.

(c) Generate 10,000 uniformly distributed random numbers on interval [0, 1). Use hist in MATLAB or

matplotlib.pyplot.hist Python to generate a histogram of all the random numbers.

Please insert your code / solution after this page.

c 2020 Stanley Chan. All Rights Reserved. 7

Exercise 10.

A collection of letters, a-z, is mixed in a jar. Two letters are drawn at random, one after the other.

(a) What is the probability of drawing a vowel (a,e,i,o,u) and a consonant in either order?

(b) Write a MATLAB / Python program to verify your answer in part (a). That is, randomly draw two

letters without replacement and check whether one is a vowel and the other is a consonant. Compute

the probability by repeating the experiment for 10000 times.

Please write your hand-written solution here.

Please insert your code / solution after this page.

c 2020 Stanley Chan. All Rights Reserved. 8

Exercise 11.

There are 50 students in a classroom.

(a) What is the probability that there is at least one pair of students having the same birthday? Show

your steps.

(b) Write a MATLAB / Python program to simulate the event, and verify your answer in (a). Hint: You

probably need to repeat the simulation for many times to obtain a probability. Submit your code and

result.

You may assume that a year only has 365 days. You may also assume that all days have equal likelihood to

be taken.

Please write your hand-written solution here.

Please insert your code / solution after this page.

c 2020 Stanley Chan. All Rights Reserved. 9