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September 12th, 2010, 02:48 PM  #1 
Member Joined: Mar 2010 Posts: 51 Thanks: 0  Correlation with Cards
Problem: Consider an ordinary deck of 52 cards, which has four suits and thirteen values. Supposed 1 card is drawn from the deck. X is the number of aces, and Y is the number of spades. Show that X and Y are uncorrelated. Are X and Y independent? End Problem Ok, I know the following probabilities: P(X=0) = 48/52 P(X=1) = 4/52 P(Y=0) = 39/52 P(Y=1) = 13/52 E(X) = 4/52 E(Y) = 13/52 I need to show that the Cov(X,Y) = 0 for them to be uncorrelated, but I'm not doing something right when calculating E(XY) We know that the Cov(X,Y) = E(XY)  E(X)E(Y) So, what is the right way to calculate this? 
September 12th, 2010, 03:33 PM  #2 
Senior Member Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3  Re: Correlation with Cards
XY can only be 1 if you draw the ace of spades. Otherwise it will be 0, as either X or Y will be 0.

September 12th, 2010, 03:36 PM  #3 
Member Joined: Mar 2010 Posts: 51 Thanks: 0  Re: Correlation with Cards
I understand the reasoning behind that. So, with the Expectation, we're really only looking at (because the rest will be multiplied by zero), E(XY) = (1)(1)(what probability)(what probability)? For them to be uncorrelated (which I know they are), E(XY) must be equal to 1. 
September 13th, 2010, 03:54 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,660 Thanks: 648  Re: Correlation with Cards
E(XY)=1/52, E(X)E(Y) = (4/52)(13/52)=1/52.

September 21st, 2010, 12:15 PM  #5 
Member Joined: Mar 2010 Posts: 51 Thanks: 0  Re: Correlation with Cards
Wow, it took me a while to realize I wasn't multiplying the denominators in the E(X)E(Y). You'll have those days sometimes. 

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