Transcript discrete random variable

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
size
x = Number of dependents in
family reported on tax return
Discrete
Distance from
home to store
x = Distance in miles from
home to the store site
Continuous
Own dog
or cat
x = 1 if own no pet;
= 2 if own dog(s) only;
= 3 if own cat(s) only;
= 4 if own dog(s) and cat(s)
Discrete
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
P(X=x)
1/
4
Probability of the
event X being ‘x’
1/
4
1
2
+ 1/4
1/
4
= 1/2
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 :-
E( X )   x P( X  x)
Expectation - example
The expected value of X is :Number of heads
(x)
0
P(X=x)
1/
4
Probability of the
event X being ‘x’
E( X )   x P( X  x)
1/
4
1
2
+ 1/4
1/
4
= 1/2
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”