Suppose that 3 balls are randomly selected from an urn containing 3 red, 4 white, and 5 blue balls. If we let X and Y denote, respectively, the number of red and white balls chosen, find the joint probability mass function and probability mass functions for X and Y , respectively.
I have the PMF for X and Y independently, but I tried checking my joint PMF response on Wolfram and it refuses to return an answer. Can someone verify if my response for the joint PMF is on the right path, or show me a better way.
$$f_X(x) = \frac{{3 \choose x}{9 \choose 3-x}}{12 \choose 3} $$ $$F_X(x) = \frac{\sum_{x=0}^n{3 \choose x}{9 \choose 3-x}}{12 \choose 3} $$
$$f_Y(y) = \frac{{4 \choose y}{8 \choose 3-y}}{12 \choose 3}$$ $$F_Y(y) = \frac{\sum_{y=0}^n{4 \choose y}{8 \choose 3-y}}{12 \choose 3} $$
$$f(x, y) = \frac{{3 \choose x}{4 \choose y}{5 \choose 3-x-y}}{12 \choose 3} $$
My general CDF solution:
$ j = x + y $ $$\ F(x, y) = \frac{\sum_{j=0}^n\sum_{x=0}^j\sum_{y=0}^{j-x}{3 \choose x}{4 \choose y}{5 \choose 3-x-y}}{12 \choose 3}$$
Without modifying it significantly, I can only find the probability the draw contains n red and/or white balls. There has to be a better, more accurate way of writing this. Looking for a response that can be used to answer a question with conditions such as the following. $$ P(a_1 < X ≤ a_2, b_1 < Y ≤ b_2) $$
Thank you in advance.