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”
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