Probability — Practice Problems

Practice problems for the Probability lecture

These practice questions accompany the Probability lecture and will be discussed in the review session.

Question 1

For the rolling of a die experiment, define \(X(\omega)= \omega\) for \(\omega\in\Omega=\{1,2,3,4,5,6\}\). Construct the CDF \(F(x)\) of \(X\).

Question 2

Suppose \(X\) is uniformly distributed on \([a,b]\): the PDF is a strictly positive constant on \([a,b]\), and is zero otherwise. Construct the CDF and PDF of \(X\).

Question 3

The joint PDF of \(X\) and \(Y\) is \(f(x,y) = ax\cdot \mathbb{1}(0<x<1,x<y<1)\), where \(\mathbb{1}(\cdot)\) is the indicator function that takes the value 1 if the condition is true and 0 otherwise.

1. Find \(a\) such that \(f(x,y)\) is a valid PDF
2. Find the marginal PDF of \(X\)
3. Find the marginal PDF of \(Y\)
4. Are \(X\) and \(Y\) independent?

Question 4

Suppose \(X\) is a Bernoulli random variable: \(\mathbb{P}[X=1]=p\), \(\mathbb{P}[X=0]=1-p\) for \(p\in(0,1)\).

1. Compute the first and second moments of \(X\)
2. Compute the variance of \(X\)

Question 5

Suppose \({\mathbb{E}}[X]={\mathbb{E}}[Y]\) for random variables \(X\) and \(Y\). Does this imply that \(X=Y\)? Prove if true or give a counterexample if false.

Question 6

For the joint PDF from Question 3, compute \({\mathbb{E}}[X]\), \({\mathbb{E}}[Y]\), \({\mathbb{E}}[XY]\), and \(\operatorname{Cov}(X,Y)\).

Question 7

Compute the quantiles of order \(\tau\) for the Uniform\((a,b)\) distribution.

Question 8

Suppose \(Y=g(X)\), where \(g\) is a strictly increasing function. Show that \(q_\tau^Y = g(q_\tau^X)\), where \(q_\tau^X\) and \(q_\tau^Y\) are the quantiles of \(X\) and \(Y\), respectively.

Question 9

Using the Markov inequality, show that \(\mathbb{P}[|X - {\mathbb{E}}[X]| \geq a] \leq \frac{\operatorname{Var}(X)}{a^2}\) for any \(a > 0\).

Question 10

The correlation coefficient \(\rho_{XY}\) between the random variables \(X\) and \(Y\) is defined as \(\rho_{XY} = \frac{\operatorname{Cov}(X,Y)}{\sqrt{\operatorname{Var}(X)\operatorname{Var}(Y)}}\). Show that \(|\rho_{XY}| \leq 1\) using the Cauchy-Schwarz inequality.

Question 11

Using Jensen’s inequality, show that if \({\mathbb{E}}[|X|^p]<\infty\) for some \(p>0\), then \({\mathbb{E}}[|X|^r]<\infty\) for all \(r\in[0,p]\). Hint: \(|x|^s\) is concave for \(0<s<1\).

Question 12

A disease affects 1% of the population: \(\mathbb{P}[D] = 0.01\). A diagnostic test has sensitivity \(\mathbb{P}[T^+ \mid D] = 0.95\) and specificity \(\mathbb{P}[T^- \mid D^c] = 0.90\). A randomly chosen person tests positive. Using the multiplication rule, the law of total probability, and Bayes’ rule, compute the posterior \(\mathbb{P}[D \mid T^+]\).

Question 13

The joint PMF of discrete \(X_1 \in \{1,2\}\) and \(X_2 \in \{1,2,3\}\) is

\(p(x_1,x_2)\)	\(X_2=1\)	\(X_2=2\)	\(X_2=3\)
\(X_1=1\)	0.10	0.15	0.05
\(X_1=2\)	0.20	0.30	0.20

Compute the marginal PMFs \(p_{X_1}(x_1)\) and \(p_{X_2}(x_2)\), and verify each sums to 1.

Question 14

Let \(X \in \{0,1\}\) and \(Y \in \{0,1,2\}\) have joint PMF

\(p(x,y)\)	\(Y=0\)	\(Y=1\)	\(Y=2\)
\(X=0\)	0.10	0.20	0.10
\(X=1\)	0.05	0.25	0.30

Compute \({\mathbb{E}}[Y \mid X=0]\) and \({\mathbb{E}}[Y \mid X=1]\), and express \({\mathbb{E}}[Y\mid X]\) as a function of \(X\).

Question 15

Prove the “taking out what is known” property for discrete random variables: for any function \(g\), \[ {\mathbb{E}}[g(X)\,Y \mid X] = g(X)\,{\mathbb{E}}[Y\mid X]. \]

Question 16

A course has two sections: A has 30 students averaging 72, and B has 20 students averaging 80. With \(S\) the section and \(G\) the grade, \({\mathbb{E}}[G\mid S=A] = 72\) and \({\mathbb{E}}[G\mid S=B] = 80\). Use the law of iterated expectations to find \({\mathbb{E}}[G]\).

Question 17

Using the joint PMF of \(X \in \{0,1\}\), \(Y \in \{0,1,2\}\) from Question 14,

\(p(x,y)\)	\(Y=0\)	\(Y=1\)	\(Y=2\)
\(X=0\)	0.10	0.20	0.10
\(X=1\)	0.05	0.25	0.30

compute \({\mathbb{E}}[Y]\) two ways and confirm they agree: (a) directly from the marginal \(p_Y\); (b) via the law of iterated expectations.

Question 18

Prove the Cauchy–Schwarz inequality: for random variables \(X, Y\) with finite second moments, \[ \big({\mathbb{E}}[XY]\big)^2 \leq {\mathbb{E}}[X^2]\,{\mathbb{E}}[Y^2]. \] Hint: consider \({\mathbb{E}}[(X - tY)^2] \geq 0\) as a quadratic in the real variable \(t\).