Discrete Random Variable

A random variable is discrete if the values it can take on within an interval is finite.

PMF and CDF

The probability mass function (PMF) describes the probability distribution over a discrete random variable.

$p(x)=P(X=x)$

The cumulative distribution function (CDF) specifies the probability of an observation being equal to or less than a given value.

$F(x)=P(X\leq x)$

We usually have tables for these in the case of discrete random variables.

Statistics

Expected value (mean):

$\mu =E(X)=\sum x_{i}P(X=x_{i})$

Distributions

Bernoulli

The bernoulli distribution describes the random variable of an experiment that has two outcomes and is performed once. The outcomes are either success or failure.

$X\sim Bernoulli(p)$

PMF

$p(1)=p,p(0)=1-p$

Statistics

$\mu =p$

$\sigma _{X}^{2}=p(1-p)$

Binomial

Repeating a bernoulli experiment $b$ times and we get a binomial random variable.

Consider an experiment with exactly two possible outcomes, conducted n times independently.

$X\sim Binomial(n,p)$

I'm sleep I'll write the details later. It should be on the equation sheet.