[Math] 5 cards / Joint Probability Function

probabilityprobability distributionsstatistics

From a deck of playing cards, you take out 5. The random variables X and Y denote the number of "aces" and "queens" in the sample, respectively. Find the joint probability function of X and Y, and their correlation coefficient.

I would like to get some help on choosing the distributions for X and Y, and how to find the joint probability distribution for X and Y from those.

Thanks!

Best Answer

You need to specify $\Pr(X=x\cap Y=y)$ for all possible pairs $(x,y)$. We can either do it by cases, or get a general formula. The possible values of $X$ range from $0$ to $4$, as do the possible values of $Y$.

There are $\dbinom{52}{5}$ ways to choose $5$ cards. All these ways are equally likely.

How many ways are there to choose $x$ Aces and $y$ Queens?

If $x+y\gt 5$, the probability that $X=x$ and $Y=y$ is clearly $0$.

Otherwise, we need to choose $x$ Aces from the $4$ available, and $y$ Queens from the $4$ available, and $5-x-y$ "other" cards from the $44$ that are neither Ace nor Queen. This can be done in $$\binom{4}{x}\binom{4}{y}\binom{44}{5-x-y}$$ ways.

For the joint distribution function, divide by $\dbinom{52}{5}$.