1. Probability mass funcitons

a jury of 12 people is randomly selected from a group of 8 red hat, 7 blue hat,3 green hat and 20 yellow hat potential jurors. Let R, B, G and Y be the # of jurors, respectively. Calculate the joit probability mass function of R,B,G,Y and the marginal probability mass function of R

2. Originally Posted by Andreamet
a jury of 12 people is randomly selected from a group of 8 red hat, 7 blue hat,3 green hat and 20 yellow hat potential jurors. Let R, B, G and Y be the # of jurors, respectively. Calculate the joit probability mass function of R,B,G,Y and the marginal probability mass function of R
You're dealing with a multinomial distribution with n = 12, $\displaystyle p_R = \frac{8}{38} = \frac{4}{19}$, $\displaystyle p_B = \frac{7}{38}$, $\displaystyle p_G = \frac{3}{38}$ and $\displaystyle p_Y = \frac{20}{38} = \frac{10}{19}$.

The joint pmf for R, B, G and Y is therefore given by

$\displaystyle p(r, b, g, y) = \frac{12!}{r!\, b! \, g! \, y!} \, \left( \frac{4}{19} \right)^r \, \left( \frac{7}{38} \right)^b \, \left( \frac{3}{38} \right)^g \, \left( \frac{10}{19} \right)^y$

where r + b + g + y = 12.

The marginal pmf of R is found by calculating

$\displaystyle \Pr(R = r) = \sum_b \sum_g \sum_y p(r, b, g, y)$

$\displaystyle = \sum_b \sum_g \sum_y \frac{12!}{r!\, b! \, g! \, y!} \, \left( \frac{4}{19} \right)^r \, \left( \frac{7}{38} \right)^b \, \left( \frac{3}{38} \right)^g \, \left( \frac{10}{19} \right)^y$

where b + g + y = 12 - r.

3. Originally Posted by mr fantastic
[snip]
The marginal pmf of R is found by calculating

$\displaystyle \Pr(R = r) = \sum_b \sum_g \sum_y p(r, b, g, y)$

$\displaystyle = \sum_b \sum_g \sum_y \frac{12!}{r!\, b! \, g! \, y!} \, \left( \frac{4}{19} \right)^r \, \left( \frac{7}{38} \right)^b \, \left( \frac{3}{38} \right)^g \, \left( \frac{10}{19} \right)^y$

where b + g + y = 12 - r.
Well, I suppose to ease my conscience I should point out that this marginal pmf is binomial .....

As you select hats you'll have successes and failures with selecting red ones. It's not hard to see then that

R ~ Binomial $\displaystyle \left( n = 12, \, p = \frac{4}{19} \right)$, that is,

$\displaystyle \Pr(R = r) = {12 \choose r} \left( \frac{4}{19}\right)^r \left( \frac{15}{19}\right)^{12-r}$.

4. Originally Posted by Andreamet
a jury of 12 people is randomly selected from a group of 8 red hat, 7 blue hat,3 green hat and 20 yellow hat potential jurors. Let R, B, G and Y be the # of jurors, respectively. Calculate the joit probability mass function of R,B,G,Y and the marginal probability mass function of R
Since the jurors are chosen (I assume) without replacement, the distribution will actually be the multivariate hypergeometric distribution:

$\displaystyle p(r, b, g, y) = \frac{ {8 \choose r} {7 \choose b} {3 \choose g} {20 \choose y}}{{ 38 \choose 12}}$

where r + b + g + y = 12.

You could also build in the r + b + g + y = 12 restriction and write the pmf as:

$\displaystyle p(r, b, g, y) = \frac{ {8 \choose r} {7 \choose b} {3 \choose g} {20 \choose {12 - r - b - g} }}{{ 38 \choose 12}}$.

The marginal pmf for red will be a hypergeometric distribution:

$\displaystyle \Pr (R = r) = \frac{ {8 \choose r} {30 \choose {12 - r}} }{{38 \choose 12}}$.

Sorry for the original mistake in interpretation