# Thread: joint probability mass function

1. ## joint probability mass function

from a deck of 52 cards,,, 13 are chosen at random. Calculate the joint probability mass function of the #'s of hearts, clubs, diamonds and spades that are selected.

2. I think this is a bivariate statistic

Originally Posted by Andreamet
from a deck of 52 cards,,, 13 are chosen at random. Calculate the joint probability mass function of the #'s of hearts, clubs, diamonds and spades that are selected.

3. Originally Posted by Andreamet
from a deck of 52 cards,,, 13 are chosen at random. Calculate the joint probability mass function of the #'s of hearts, clubs, diamonds and spades that are selected.
Read this thread - this time you have a multinomial distribution with n = 13 and pr(heart) = pr(spade) = pr(diamond) = pr(club) = 1/4.

4. Originally Posted by Andreamet
from a deck of 52 cards,,, 13 are chosen at random. Calculate the joint probability mass function of the #'s of hearts, clubs, diamonds and spades that are selected.
If the choosing is done without replacement, the pmf will be a multivariate hypergeometric distribution:

$p(h, d, c, s) = \frac{ {13 \choose h} {13 \choose d} {13 \choose c} {13 \choose s}}{ {52 \choose 13} }$

where h + d + c + s = 13.

You could also build in the h + d + c + s = 13 restriction and write it as:

$p(h, d, c, s) = \frac{ {13 \choose h} {13 \choose d} {13 \choose c} {13 \choose {13 - h - d - c}}}{ {52 \choose 13} }$.