Random Variables

1. Random Variables A random variable is a rule that assigns exactly one value to each point in a sample space for an experiment. A random variable can be classified as being either discrete or continuous depending on the numerical values it assumes. A discrete random variable may assume either a finite number of values or an infinite sequence of values. A continuous random variable may assume any numerical value in an interval or collection of intervals.

2. Random Variables Question Random Variable x Type Family x = Number of dependents in Discrete size family reported on tax return Distance from x = Distance in miles from Continuous home to store home to the store site Own dog x = 1 if own no pet; Discrete or cat = 2 if own dog(s) only; = 3 if own cat(s) only; = 4 if own dog(s) and cat(s)

3. Discrete Probability Distributions The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable. The probability distribution is defined by a probability function, denoted by P(x), which provides the probability for each value of the random variable.

4. Discrete Probability Distributions Two Basic Properties of a Discrete Probability Distribution: P(x) > 0 for each value of x. 2. ΣP(x) = 1. We can describe a discrete probability distribution with a table, graph, or equation.

5. Random Variables Mean, Variance, and Standard Deviation The mean, or expected value, of the discrete random variable x with probability distribution P(x) is a measure of its central location.  = xP(x) The variance summarizes the variability in the values of a random variable.  2 = (x - )2P(x) The standard deviation, , is defined as the positive square root of the variance.

6. Random Variables Random Variables Mean, Variance, and Standard Deviation Shortcut Formula for the Variance of a Random Variable:  2 = x 2P(x) -  2