Discrete Random Variables A random variable is a function that assigns

a numerical value to each simple event in a sample space. Range – the set of real numbers Domain – a sample space from a random

experiment A discrete random variable can assume only

a countable (finite or countably infinite) number of values.

A continuous random variable can assume an uncountable number of values

Counting numbers The values of a discrete random variable

are countable I.e. they can be paired with the counting numbers 1,2, …

Counting numbers, 0, the negatives of counting numbers, and the ratios of counting numbers and their negatives (rational numbers) are inadequate for measuring.

Consider the square root of 2, the length of the diagonal of a square of side 1.

Measuring Numbers The values of a continuous random

variable are uncountable, and hence resemble the numbers comprising a continuum or interval, needed for measuring

Measurements are always made to an interval, however small.

Mass functions vs. density functions With discrete random variables,

probabilities are for ‘discrete’ points Probability functions of discrete random

variables are called probability mass functions

With continuous random variables, probabilities are for intervals

Probability functions of continuous random variables are called probability density functions

Expected value of a discrete random variable

E(X) = {x*[P(X=x)]}={x*p(x)} =

Var(X) = {(x-)2 * [P(X=x)]} = {(x-2*p(x)} = 2

Laws of Expected Value

E( c ) = cE ( cX) = cE(X)E(X+Y) = E(X) + E(Y)E(X - Y) = E(X) – E(Y)E(X*Y) + E(X) * E(Y) if and only of X

and Y are independent

Laws of Variance

V ( c ) = 0V(cX) = c2*V(X)V(X+c) = V(X)V(X+Y) = V(X) + V(Y) if and only if X

and Y are independentV(X – Y) = V(X) + V(Y) if and only if X

and Y are independent