A random variable is a rule that assigns a numerical value to each outcome of an experiment. Random variables can be either discrete or continuous. A discrete random variable may assume countable values, while a continuous random variable can assume any value in an interval. The probability distribution of a random variable describes the probabilities of the variable assuming different values. For a continuous random variable, the probability of it assuming a value within an interval is given by the area under the probability density function within that interval.

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. 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 which provides the probability for each value of the random variable.

4. Continuous Probability Distributions A continuous random variable can assume any value in an interval on the real line or in a collection of intervals. It is not possible to talk about the probability of the random variable assuming a particular value. Instead, we talk about the probability of the random variable assuming a value within a given interval.

5. The probability of the random variable assuming a value within some given interval from x 1 to x 2 is defined to be the area under the graph of the probability density function between x 1 and x 2 . Continuous Probability Distributions

6. Probability Density Function For a continuous random variable X, a probability density function is a function such that (1) (2) (3)