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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(1) = p, p(0) = 1 - p }

Statistics

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu = p }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma^2_X = p (1 - p) }

Binomial

Repeating a bernoulli experiment Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} times and we get a binomial random variable.

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \sim Binomial(n, p) }

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