Probability Distributions - Discrete Random Variables Outcomes and Events

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.

Random Variables QuestionRandom Variable x Type Family x = Number of dependents in Discrete size family reported on tax return Distance from x = Distance in miles fromContinuous 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)

Probability Distributions The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable. 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) P(X=x) 1 / 4 1 / / 4 1 / 4 = 1 / 2 Probability of the event X being ‘x’

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

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