- #1
jimholt
- 12
- 0
Obtain marginal probability mass function (pmf) given joint pmf
Not really a homework question, but it does have a homeworky flavor, doesn't it...
Homework Statement
Given a join probability mass function of two variables, is it always possible to obtain the marginals?
E.g., if I have a joint mass function for two Bernoulli random variables X and Y, like this:
[tex]
f(x,y) = \begin{cases} 1/2 & \mbox{if } (x,y) = (0, 1) \\
1/2 & \mbox{if } (x,y) = (1, 0) \\
0 & \mbox{otherwise} \end{cases}
[/tex]
Can I obtain the marginals for X and Y?
The attempt at a solution
I want to say yes, but if the marginals for X and Y are
[tex]
f(x) = \begin{cases} 1/2 & \mbox{if } x = 0 \\
1/2 & \mbox{if } x = 1 \\
0 & \mbox{otherwise} \end{cases}
[/tex]
and
[tex]
f(y) = \begin{cases} 1/2 & \mbox{if } y = 0 \\
1/2 & \mbox{if } y = 1 \\
0 & \mbox{otherwise} \end{cases}
[/tex]
Then that produces a joint mass function
[tex]
f(x,y) = \begin{cases} 1/4 & \mbox{if } (x,y) \in \{(0, 0), (0, 1), (1, 0), (1, 1) \} \\ 0 & \mbox{otherwise} \end{cases}
[/tex]
which is clearly wrong.
So what's the right way to get at the marginals, assuming they, er, exist?
Not really a homework question, but it does have a homeworky flavor, doesn't it...
Homework Statement
Given a join probability mass function of two variables, is it always possible to obtain the marginals?
E.g., if I have a joint mass function for two Bernoulli random variables X and Y, like this:
[tex]
f(x,y) = \begin{cases} 1/2 & \mbox{if } (x,y) = (0, 1) \\
1/2 & \mbox{if } (x,y) = (1, 0) \\
0 & \mbox{otherwise} \end{cases}
[/tex]
Can I obtain the marginals for X and Y?
The attempt at a solution
I want to say yes, but if the marginals for X and Y are
[tex]
f(x) = \begin{cases} 1/2 & \mbox{if } x = 0 \\
1/2 & \mbox{if } x = 1 \\
0 & \mbox{otherwise} \end{cases}
[/tex]
and
[tex]
f(y) = \begin{cases} 1/2 & \mbox{if } y = 0 \\
1/2 & \mbox{if } y = 1 \\
0 & \mbox{otherwise} \end{cases}
[/tex]
Then that produces a joint mass function
[tex]
f(x,y) = \begin{cases} 1/4 & \mbox{if } (x,y) \in \{(0, 0), (0, 1), (1, 0), (1, 1) \} \\ 0 & \mbox{otherwise} \end{cases}
[/tex]
which is clearly wrong.
So what's the right way to get at the marginals, assuming they, er, exist?
Last edited: