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Index > Fundamentals of probability

Probability questions

This page collects 200 questions about probability that you can use to test your preparation. Read the questions and for each one of them ask yourself whether you would be able to answer. If you think you do not know how to answer, you can follow the links at the end of each section and revise the relevant concepts. Each section contains a group of more difficult questions (advanced questions). You can safely skip them if you are taking a first course in probability.

Probability exercises. Are you looking for solved probability exercises? There are hundreds of problem sets on Statlect. At the end of each of the sections below you can find links to lectures that contain exercises with solutions.

Table of Contents

Probability and events

More advanced questions:

Lecture where you can find the answers Number of solved exercises in the lecture
Probability 3
Zero-probability events 2

Conditional probability and independence

More advanced questions:

Lecture where you can find the answers Number of solved exercises in the lecture
Conditional probability 3
Bayes' rule 3
Independent events 3

Random variables

Plot of the pdf of a normal distribution  to which the questions above refer

More advanced questions:

Lecture where you can find the answers Number of solved exercises in the lecture
Random variables 6
Probability mass function No problem sets / 1 example
Probability density function No problem sets / 1 example
Cumulative distribution function No problem sets / 1 example

Random vectors

More advanced questions:

Lecture where you can find the answers Number of solved exercises in the lecture
Random vectors 6
Conditional probability distributions 3
Independent random variables 3

Expected value

More advanced questions:

Lecture where you can find the answers Number of solved exercises in the lecture
Expected value 6
Properties of the expected value 3
Conditional expectation 3
Expected value and the Lebesgue integral (advanced) N/A

Variance and other moments

More advanced questions:

Lecture where you can find the answers Number of solved exercises in the lecture
Variance 6
Covariance matrix 3
Moments No problem sets / 2 examples
Cross-moments No problem sets / 2 examples

Covariance and correlation

More advanced questions:

Lecture where you can find the answers Number of solved exercises in the lecture
Covariance 6
Linear correlation 3
Independent random variables 3

Inequalities

More advanced questions:

Lecture where you can find the answers Number of solved exercises in the lecture
Markov's inequality 1
Chebyshev's inequality 1
Jensen's inequality 1

Transformations of random variables

More advanced questions:

Lecture where you can find the answers Number of solved exercises in the lecture
Functions of random variables 3
Functions of random vectors 2
Sums of independent random variables 2

Moment generating and characteristic functions

More advanced questions:

Lecture where you can find the answers Number of solved exercises in the lecture
Moment generating function 3
Joint moment generating function 3
Characteristic function 3
Joint characteristic function 3
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