# Distribution of card hands

1. Nov 3, 2012

### nikki92

X = # of clubs in a 13 card hand drawn at random without replacement from 52 card deck
Y= # of queens in the same 13 card hand

the pdf of x is f_x (x) = (13 choose x)( 39 choose 13-x) / (52 choose 13) for x=< x =< 13
and o otherwise

the joint distribution of x and y = (13 choose x) ( 5 choose y) ( 34 choose 13-x-y)/ (52 choose 13) for 0 =<x =<13 0 =< y =< 4 0<=x+y<=13 and 0 otherwise

What is E(Y) and what is the covariance matrix (X,Y)?

E[Y] = Sum y=1 to 4 of y* (4 choose y)(48 choose 13-y)/ (52 choose 13) ?

I have no idea on the matrix.

2. Nov 4, 2012

### chiro

Hey nikki92.

The covariance of two random variables is given by Cov(X,Y) = E[XY] - E[X]E[Y] where E[XY] = Summation (over x) Summation over y P(X=x,Y=y)*xy (Or an integral for continuous random variable).

The covariance matrix has Cov(Xi,Xj) at the (i,j) position in the matrix and note that Cov(Xi,Xi) = E[Xi^2] - {E[Xi]}^2 = Var[Xi] so you will have four entries with Var[X1], Var[X2] in (1,1) (2,2) and Cov(X,Y) in the other positions.