University of California, Los Angeles
Department of Statistics
Statistics 100A
Instructor: Nicolas Christou
Joint probability distributions
So far we have considered only distributions with one random variable.
There are many
problems that involve two or more random variables. We need to examine how these ran
dom variables are distributed together (“jointly”). There are discrete and continuous joint
distributions.
Discrete:
Here is an Example:
Let
X
be the number of puppies born, and
Y
be the number of puppies survived for a certain
breed of dog. Suppose the following table describes the distribution of
X
and
Y
:
X
3
4
5
0
0.31
0.21
0.21
0.73
Y
1
0.03
0.04
0.05
0.12
2
0.02
0.03
0.04
0.09
3
0.01
0.02
0.03
0.06
0.37
0.30
0.33
1.0
Notation:
•
Joint (or bivariate) probability distribution of
X
and
Y
:
f
XY
(
x, y
) =
P
(
X
=
x, Y
=
y
)
Example:
•
Always
X
x
X
y
f
XY
(
x, y
) = 1
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
•
Marginal distribution of
X
:
f
X
(
x
) =
X
y
f
XY
(
x, y
)
Example:
•
Marginal distribution of
Y
:
f
Y
(
y
) =
X
x
f
XY
(
x, y
)
Example:
•
Conditional probability function:
f
Y

X
(
y

x
) =
P
(
Y
=
y

X
=
x
) =
P
(
Y
=
y, X
=
x
)
P
(
X
=
x
)
Or
f
Y

X
(
y

x
) =
f
XY
(
x, y
)
f
X
(
x
)
Example:
•
Joint cumulative distribution function:
F
XY
(
x, y
) =
P
(
X
≤
x, Y
≤
y
)
Example:
•
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 Wu
 Statistics, Normal Distribution, Probability, Probability theory, probability density function

Click to edit the document details