Tag Archives: binomial distribution

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 […]

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 […]

Where does the Poisson distribution come from?

The Poisson distribution is a discrete probability distribution on the natural numbers $0,1,2,\dots$. Its density function depends on one parameter $\mu$ and is given by $$d(n) = \frac{e^{-\mu}\mu^n}{n!}$$ Not surprisingly, the parameter $\mu$ is the mean, which follows from the exponential series $$e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$$ Here is what the density function looks like when […]