Description
problem 1. Let X be a random variable with expectation µ and standard deviation σ. Find each of the
following, expressed in terms of µ and σ:
(a) Ex(2X
2 − 3X + 4);
(b) Var(−3X + 4).
Solution:
problem 2. Let X be a continuous random variable with the following PDF, fX(x), where c is a constant:
fX(x) = (
cx(1 − x)
2
if x ∈ (1, 2)
0 otherwise
(a) Find the value of c.
(b) Find Ex(X).
(c) Find Var(X).
Solution:
problem 3. Recall that a random variable with range {0, 1} is called an indicator random variable. Let
X and Y be two independent indicator random variables, with Pr(X = 1) = p and Pr(Y = 1) = r. Find
Ex((X − Y)
2
) in terms of p and r.
Solution:
problem 4. There are n people on a friendship-matchmaking app. We assume that each pair of people
will connect and become friends with probability 0.1, independently of other pairs.
(a) You are one of the n people on the app. Let X be the random variable denoting the number of
friends you will make on the app. Find the PDF of X.
(b) What is the expected number of friends you will make on the app?
(c) What is the probability that you will make at least 2 friends on the app?
(d) What is the expected number of friendships formed on the app, i.e. what is the expected number of
pairs that will connect and become friends?
Solution:
problem 5 (Problem 19.33 in the textbook). Alice has decided to volunteer in a bakery. She has taken
CS 237 and she knows that it will be great fun if she uses the following strategy for deciding how many
loaves of bread to bake. Each day, Alice first rolls a fair, k-sided die to get a number n ∈ {1, 2, . . . , k}, and
then she tosses a fair coin. Alice bakes n loaves of bread if the coin is heads, and she bakes 2n loaves if
the coin is tails.
(a) For any positive integer ` ≤ 2k, what is the probability that Alice bakes exactly ` loaves of bread on
any given day?
Hint: you can express your solution by cases.
(b) What is the expected number of loaves that Alice bakes on any given day?
(c) Suppose Alice volunteers in the bakery every day for 30 days. What is the expected total number of
loaves that Alice bakes?
You may find the following identity helpful:
n
∑
i=1
i =
n(n + 1)
2
Solution:
problem 6 (Simulations). Download the HW9 Jupyter notebook (notebook coming soon!). Complete all
the exercises in the notebook. Submit the Jupyter notebook with your solutions to the Homework 9
Programming assignment on Gradescope. Your submission should be a single .ipynb file.
H9-2
