(adsbygoogle = window.adsbygoogle || []).push({}); Obtain marginal probability mass function (pmf) given joint pmf

Not really a homework question, but it does have a homeworky flavor, doesn't it...

The problem statement, all variables and given/known data

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?

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# Always possible to obtain marginals from joint pmf?

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