5 Random Variables and Distributions

random variable

A random variable is a function

X : Ω → R

that has a (well-defined) cumulative distribution function, FX : R → [0, 1],

defined as

FX (x) := P({X ≤ x}), for all x ∈ R.

The set of values that the function X can take is called its range and is

denoted by RX .

Cumulative distribution

Fx

Probability x is less than or equal to a given value

discrete random variable

random variable X

which can assume at most countably many values is called a discrete

random variable

probability mass function for a discrete random variabl x

For a discrete random

variable X, the probability mass function (pmf) is

pX (x) := P({X = x}) for all x ∈ R

if 2 discrete random variable x and y have the same distribution then

they have the same pmf

X ∼ Y (“X distributed like Y ”) if pX (x) = pY (x) for all x ∈ R

joint probability mass function

pX,Y (x, y) = P({X = x} ∩ {Y = y}) for x, y ∈ R

Px(x)

pY(y)

joint cumulative distribution function

when are discrete random variable indepedenet

pX,Y (x, y) = pX (x) × pY (y) for all x, y ∈ R.

when is a family of discrete random variables independent

The graph of the pmf of the sum of two die roll

cumulative distibution function

robability of being less than or equal to that value