Chapter 16

Random Variable

A random variable assumes any of several different values as a result of some random event. Random Variables are denoted by a capital letter such as X.

Continuous Random Variable

A random variable that can take any numeric value within a range of values is called a continuous random variable. The range may be infinite or bounded at either or both ends.

Discrete Random Variable

A random variable that can take on of a finite number* of distinct outcomes is called a discrete random variable * Technically, there could be an infinite number of outcomes as long as they are countable. Essentially that means we can imagine listing them all in order, like the counting numbers 1,2,3,4,5...

Adding or Subtracting Random Variables

E(X+Y) = E(X) + E(Y) or E(X-Y) = E(X) - E(Y) and if X and Y are independent, Var(X+Y) = Var(X) + Var(Y) or Var(X-Y) = Var(X) + Var(Y). (The Pythagorean Theorem of Statistics).

Changing a Random Variable by a Constant

E(X+c) = E(X) + c Var(X+c) = Var(X) E(X-c) = E(X) - c Var (X-c) = Var(X) E(aX) = aE(X) Var(aX) = a^2*Var(X)

Expected Value

The expected value of a random variable is its theoretical long run average value, the center of its model. Denoted μ or E(X), it is found (if the random variable is discrete) by summing the products of variable values and probabilities. μ = E(X) = Σx*P(x).

Probability Model

The probability model is a function that associates a probability P with each value of a discrete random variable X, denoted P(X=x), or with any interval of values of a continuous random variable.

Standard Deviation

The standard deviation of a random variable describes the spread in the model and is the square root of the variance. σ = SD(X) = VAR(X)^(1/2)

Variance

The variance of a random variable is the expected value of the squared deviation from the mean. For discrete random variables they can be calculated as: σ^2 = Var(X) = Σ(x-μ)^2*P(x)