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

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)

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) 0 1 2

P(X=x) 1/4 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)

)()( xXPxXE

The expected value of X is :-

Expectation - example

)()( xXPxXEThe expected value of X is :-

Number of heads(x) 0 1 2

P(X=x) 1/4 1/4 + 1/4 1/4

= 1/2

Probability of the event X being ‘x’

E(X) = 0 x 1/4 + 1 x 1/2 + 2 x 1/4

= 1 “You would expect 1 head out of every 2 throws”