1
$\begingroup$

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!

$\endgroup$
1
$\begingroup$

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 $\dbinom{52}{5}$.

$\endgroup$
  • $\begingroup$ I appreciate your clear answer! Thanks! $\endgroup$ – user61652 Feb 10 '13 at 1:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.