Hi all, I would be grateful for some help in finishing this exercise. I've taken care of the first part myself, I just got stuck in the second. Here goes:
Five cards are drawn at random and without replacement from an ordinary deck of cards. Let x and y denote respectively the number of spades and the number of hearts that appear in five cards.
1) Determine the joint pmf of x and y
2) Find the marginal pmfs
3) What is the conditional pmf of y for given x?
In 1), it is easy to see that the joint pmf is given by: \(\displaystyle \displaystyle \frac{\binom{13}{x}\binom{13}{y}\binom{26}{5-x-y}}{\binom{52}{5}}\) with \(\displaystyle \displaystyle x+y\leq5\), both x and y integers. No problems there.
In 2) however, the problem is that I do not know how to get neat expressions for the marginal distributions. I know that in order to find the marginal distribution of each variable you have to sum the joint pmf over the support of the other variable, i.e. for x, \(\displaystyle \displaystyle p(x)= \frac{\tbinom{13}{x}}{\tbinom{52}{5}}\sum_{y=0}^{5-x}\binom{13}{y}\binom{26}{5-x-y}\) and similarly for y, \(\displaystyle \displaystyle p(y)= \frac{\binom{13}{y}}{\binom{52}{5}}\sum_{x=0}^{5} \binom{13}{x}\binom{26}{5-x-y}\). Of course the upper bounds of summation can be interchanged.
Without a neat expression for the marginal distributions, I don't think you can get an expression for the conditional pmf in 3).
In any case for 3) as solution the book gives \(\displaystyle \displaystyle \frac{\binom{13}{y}\binom{26}{5-x-y}}{\binom{39}{5-x}}\) while what I can get after dividing the joint pmf with the marginal pmf of x and cancelling is,\(\displaystyle \displaystyle \frac{\binom{13}{y}\binom{26}{5-x-y}}{\sum_{y=0}^{5-x}\binom{13}{y}\binom{26}{5-x-y}}\)
Any suggestion as to how I can get there? Thanks in advance.
Five cards are drawn at random and without replacement from an ordinary deck of cards. Let x and y denote respectively the number of spades and the number of hearts that appear in five cards.
1) Determine the joint pmf of x and y
2) Find the marginal pmfs
3) What is the conditional pmf of y for given x?
In 1), it is easy to see that the joint pmf is given by: \(\displaystyle \displaystyle \frac{\binom{13}{x}\binom{13}{y}\binom{26}{5-x-y}}{\binom{52}{5}}\) with \(\displaystyle \displaystyle x+y\leq5\), both x and y integers. No problems there.
In 2) however, the problem is that I do not know how to get neat expressions for the marginal distributions. I know that in order to find the marginal distribution of each variable you have to sum the joint pmf over the support of the other variable, i.e. for x, \(\displaystyle \displaystyle p(x)= \frac{\tbinom{13}{x}}{\tbinom{52}{5}}\sum_{y=0}^{5-x}\binom{13}{y}\binom{26}{5-x-y}\) and similarly for y, \(\displaystyle \displaystyle p(y)= \frac{\binom{13}{y}}{\binom{52}{5}}\sum_{x=0}^{5} \binom{13}{x}\binom{26}{5-x-y}\). Of course the upper bounds of summation can be interchanged.
Without a neat expression for the marginal distributions, I don't think you can get an expression for the conditional pmf in 3).
In any case for 3) as solution the book gives \(\displaystyle \displaystyle \frac{\binom{13}{y}\binom{26}{5-x-y}}{\binom{39}{5-x}}\) while what I can get after dividing the joint pmf with the marginal pmf of x and cancelling is,\(\displaystyle \displaystyle \frac{\binom{13}{y}\binom{26}{5-x-y}}{\sum_{y=0}^{5-x}\binom{13}{y}\binom{26}{5-x-y}}\)
Any suggestion as to how I can get there? Thanks in advance.