Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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!

share|improve this question

1 Answer 1

up vote 1 down vote accepted

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 $\dbinom{52}{5}$.

share|improve this answer
    
I appreciate your clear answer! Thanks! –  user61652 Feb 10 '13 at 1:58

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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