Probability Theory is the branch of mathematics that deals with uncertainty, underpinning all of statistics and machine learning.

Probability Interview

Introduction

Probability Axioms

Implications of the Probability Axioms

Conditional Probability

Bayes' Rule

Combinatorics

Lazy Raters

Fake Algorithm Reviews

Three Zebras

Poker Pair

Basic Probability

All areas of study in math can roughly be divided into two camps: discrete mathematics and continuous mathematics. Perhaps the best way to describe the difference between the two is to talk about what each of the branches means by "number."

Discrete Random Variables

Expected Values

Variance

Discrete Uniform Distributions

Bernoulli and Binomial Distributions

Geometric Distributions

Poisson Distributions

N Dice

Fair Coin

Converted Sessions

Biased Five Out of Six

Discrete Distributions

Continuous probability distribution: A probability distribution in which the random variable X can take on any value (is continuous).

Continuous Random Variables

The Normal Distribution, From Scratch

2X - Y

Non-normal Probability Distribution

Swimmer Survival

Continuous Distributions

Multivariate distributions show comparisons between two or more measurements and the relationships among them. For each univariate distribution with one random variable, there is a more general multivariate distribution.

Joint Distributions & Expectations

Conditional Distributions & Expectation

Covariance

Multinomial Distributions

Multivariate Normal Distributions

Independence of Random Variables

Covariance vs. Correlation

Random Seed Function

Multivariate Distributions

Thus far in this course, we have considered random variables under an idealized scenario where we know the distribution of the random variable.

Sample vs. Population

Convergence Theorems of Samples

500 Cards

First to Six

Unbiased Estimator

Ad Raters: Part 2

Sampling Theorems

Let's tackle harder probability questions you'd expect to see from quantitative finance or trading interviews.

Jars and Coins

Estimated Rounds

Secret Wins

HHT or HTT

Flipping 576 Times

Probability Questions: Hard

Probability theory underlies much of what we do as data scientists. Learn the basics and then move on to intermediate applications. 

Understand core probability concepts and theory across different distributions

<h2>Discrete case</h2>

A discrete bivariate probability mass function (pmf) is a function \(f_{X,Y}(x,y):\mathbb{Z}\times\mathbb{Z}\rightarrow[0,1]\) such that:

<ol>
<li>\(f_{X,Y}(x,y)\geq0 \ \forall \ (x,y)\in\mathbb{Z}\times\mathbb{Z}\)</li>

<li>\[\sum_{x\in\mathbb{Z}}\sum_{y\in\mathbb{Z}}f_{X,Y}(x,y)=1\]</li>
</ol>

Joint pmfs have the same meaning in terms of random variables as univariate pmfs:

\[\mathbb{P}(X=x,Y=y)=f_{X,Y}(x,y)\]

We can generalize this to more than two random variables by considering an \(n\)-dimensional vector of random variables (a “random vector”, if you will), say \(\Xi=[X_1,X_2,\dots,X_n]^T\). If \(\mathbf{x}=[x_1,x_2,\dots,x_n]^T\in\mathbb{Z}^n\) then

Probability Interview

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