Discrete Random Variables

1. 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

2. 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.

3. 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.

4. 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

5. Expected value of a discrete random variable • E(X) = S {x*[P(X=x)]}=S{x*p(x)} = m • Var(X) = S {(x-m)2 *[P(X=x)]} = S {(x-m)2*p(x)} = s2

6. Laws of Expected Value E( c ) = c E ( 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

7. Laws of Variance V ( c ) = 0 V(cX) = c2*V(X) V(X+c) = V(X) V(X+Y) = V(X) + V(Y) if and only if X and Y are independent V(X – Y) = V(X) + V(Y) if and only if X and Y are independent