Category Archives: probability

## The binomial’s variance through generating functions

In the post Binomial distribution: mean and variance, I proved that if $X$ is a binomial random variable with parameters $n$ (trials) and $p$ (probability of success) then the variance of $X$ is $np(1-p)$. If you’ll notice my proof was by induction. You might ask, why would I do that? It’s certainly one of the […]

## Inductive formula for binomial coefficients

In the last post on the mean and variance of a binomial random variable, we used the following formula: $$\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}.$$ Let’s just take a moment to prove this formula. Of course, how we prove it depends on what definition you use of the binomial coefficients. We have to start somewhere, after […]

## Binomial distribution: mean and variance

A Bernoulli random variable with parameter $p$ is a random variable that takes on the values $0$ and $1$, where $1$ happens with probability $p$ and $0$ with a probability of $1-p$. If $X_1,\dots,X_n$ are $n$ independent Bernoulli random variables, we define $$X = X_1 + \cdots + X_n.$$ The random variable $X$ is said […]

It is said that Markov originally invented Markov processes to understand how some letters follow other letters in poetry. Recall that a Markov process is a probability random process that models moving from one state to another state, where the possible states is some set. There is a fixed probability from moving from each state […]

## Cereal box prizes and transition matrices

If you don’t know what a transition matrix is, you might want to read the transition matrix post before reading this one. Transition matrices can be used to solve some classic probability problems. For example, consider the following problem: Suppose in each cereal box you buy there is one number in the set $\{1,2,3,4,5\}$. You […]

## Transition matrices

Imagine $n$ states of a system in a discrete-time stochastic system. For each pair of states $i$ and $j$, there is a probability $p_{ij}$ of moving to state $j$ in the next time step, given that the system is in state $i$. Each of these probabilities can be put in a matrix, known as the […]

## Expected iterations for a finite random walk

Consider three cells as so: A player (the blue disc) starts out in the left-most cell, and discrete time starts. At each step in time, the player has a 1/2 probability of moving left and a 1/2 probability of moving right. If the player chooses to move left but cannot because it is in the […]