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.
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(x) = P(X = x)}
The cumulative distribution function (CDF) specifies the probability of an observation being equal to or less than a given value.
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 F(x) = P(X \leq x)}
We usually have tables for these in the case of discrete random variables.
Statistics
Expected value (mean):
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 = 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.
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 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.
