Probability Distributions and Random Variables

1. Probability Distributions- Discrete Random VariablesOutcomes and Events

2. Random Variables A random variable uses 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.

3. 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)

4. E.g. Probabilities of flipping a head from 2 coin tosses X - is the random variable for the event ‘number of heads’ x - is the number of heads for the calculations Number of heads (x) 0 1 2 P(X=x) 1/41/4 + 1/4 1/4 Probability of the event X being ‘x’ = 1/2 Probability Distributions The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable.

5. The expected value of X is :- Expectation … it’s written E(X) The mean of the random variable X is called the expected value of X

6. Number of heads (x) 0 1 2 P(X=x) 1/41/4 + 1/4 1/4 Probability of the event X being ‘x’ = 1/2 Expectation - example The expected value of X is :- E(X) = 0 x1/4 + 1 x1/2 + 2 x1/4 = 1 “You would expect 1 head out of every 2 throws”