ENGRD 2700: Basic Engineering Probability and Statistics
Spring 2022
Homework 1
Due Tuesday, February 1st at 5pm. Submit solutions to Gradescope by clicking the name of the assignment.
See syllabus for detailed submission instructions.
When completing this assignment (and all subsequent ones), keep in mind the following:
• You must complete the homework individually and independently.
• Provide evidence for each of your answers. If a calculation involves only very minor computation then
explain the computation you performed and give the results. If a calculation involves more complicated
steps on many many records then hand in the calculations and formulas for the first few records only.
• Write clearly and legibly. You are encouraged to type your work although you do not have to. We may
deduct points if your answers are difficult to read or disorganized.
• For questions that you answer using Python, attach any code that you write, along with the relevant
plots.
• Submit your homework as a single pdf file on Gradescope.
1. Please answer the following questions.
(a) Why did you take this course?
(b) What major do you intend to study?
(c) What type of job would you like to get when you graduate?
(d) What salary would you like to make when you graduate?
(e) Have you heard of Operations Research before coming to Cornell? If so, where did you you hear
about it?
(f) What grade do you expect to receive in this course?
Why I am assigning this question: This is partially to get to know you.
2. You are a financial planning consultant at Cornell Federal Credit Union. A 22-year-old Cornell graduate
asks you the following question: “I would like to set up my own insurance policy by opening a trust
account into which I can make monthly payments starting now, so that upon my death or my ninetyfifth birthday—whichever comes sooner—the trust can be expected to be worth 500, 000. How much
should I invest each month?”
(a) What information do you need to solve this question?
Why I am assigning this question: This is get you out of the idea that all questions have exact
solutions. They don’t. Moreover, it is always important to know what information you will need to
know in order to solve a question. When you graduate, you will see that it is always important to
know what assumptions people are making in order to ”get an answer”. By the end of this course you
should be able to rigorously solve this question with some assumptions.
3. Compute the first and second derivative of the function f (x) where
f (x) = e−
(x−µ)2
2σ 2
.
(1)
(a) What is the critical point of the function?
The derivative of this function is
f 0 (x) = −
2
(x − µ) − (x−µ)
2σ 2
e
.
σ2
The critical points are the solution of the equation
f 0 (x) = 0.
Therefore, the critical point is x = µ.
(b) What is the point of inflection?
The points of inflections are the solutions of the equation f 00 (x) = 0. Now,
f 00 (x) =
2
(x − µ)2 − σ 2 − (x−µ)
2σ 2
.
e
σ4
Thus, f 00 vanishes when
(x − µ)2 − σ 2 = 0,
i.e x = µ ± σ.
(c) Does the first derivative yield a minimum or a maximum?
From part (a), f 0 (µ) = 0. Also, from part (b), f 00 (µ) = − σ12 e−
is the maximum.
(µ−µ)2
2σ 2
= − σ12 < 0. Hence, x = µ
Why I am assigning this question: This is get you familiar with taking derivatives of complicated
functions. Also I want you to realize that you have already done maximum likelihood estimation in
your calculus class. However, they did not tell you that you did. Moreover, the function f (x) is the
probability distribution of the normal distribution, which we will study extensively in this class.
4. Compute explicit expressions for the following sums: (no partial credit)
(a)
∞
X
ρj
0≤ρ<1
(2)
j=0
P∞
However, the answer is computed by setting the sum equal to x i.e. x = j=0 ρj . Then
multiplying the sum by ρ and noticing that ρx = x − 1. Solving for x yields the above expression.
1
1−ρ .
(b)
∞
X
ρj
0≤ρ<1
(3)
j=k
P∞
ρk
. This obtained by noticing that the above
The above sum can be written as ρk j=0 ρj = 1−ρ
k
sum is just sum in part (a) but multiplied by ρ to get the appropriate shift.
(c)
k
X
ρj
(4)
j=0
1−ρk+1
1−ρ .
However, this does not work when ρ = 1. When ρ = 1, we need to use L’hopitals rule to
get the answer k + 1.
(d)
k2
X
ρj
(5)
j=k1
The above sum equals
k2
X
ρj −
j=0
kX
1 −1
j=0
k2 +1
=
ρj
1−ρ
1−ρ
−
1 − ρk 1
ρk1 − ρk2 +1
=
.
1−ρ
1−ρ
However, this also does not work for ρ = 1. When ρ = 1 we have that the answer is k2 − k1 + 1.
Why I am assigning this question: This is to remind you about geometric sums that you
learned in high school. We will see later in the course that geometric sums are intimately linked
with geometric random variables. If you take ORIE 4130, you will see that geometric series are
important for describing how long people wait in line at RPCC for the dim-sum.
5. Show that the following double integral is equal to the harmonic series
Z
1
1
Z
n(1 − xy)n−1 dxdy = 1 +
0
0
1
1
+ · · · + ≈ log(n) + γ
2
n
where γ ≈ .5772 is the Euler-Mascheroni constant.
Why I am assigning this question: Reviewing integration and making connections to harmonic
series. You might find the answers to the previous question helpful here as there are connections to
geometric series as well.
The above integration is just
Z 1 Z
0
Z
0
1
=
0
Z
1
=
0
Z
1
=
0
=
n(1 − xy)n−1 dx dy
0
1
=
Z
1
Z
1
n(1 − z)n−1 dz dy (substituting xy = z)
y
0
Z y
1
n−1
n(1 − z)
dz dy
y
0
Z 1
1 − (1 − y)n
1
n
(1 − (1 − y) )dy =
dy
y
1 − (1 − y)
0
!
n−1
X
(1 − y)k dy (from problem 4(c))
k=0
n−1
XZ 1
k=0
y
0
(1 − y)k dy =
n−1
X
k=0
1
1
1
= 1 + + ··· + .
k+1
2
n
6. Describe the similarities and differences between the words ”data science”, ”machine learning”, ”statistics”, ”probability”, ”analytics”.
Why I am assigning this question: Everyone throws words around without knowing what they
mean. Figure out what they mean.