ECE 302: Probabilistic Methods in Electrical and Computer Engineering Homework 1 solution

Description

Exercise 1.
Calculate the infinite series
X∞
k=0
k ·

2
3
k+1
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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)
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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
.
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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