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September 12th, 2010, 01:48 PM   #1
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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?
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September 12th, 2010, 02:33 PM   #2
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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.
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September 12th, 2010, 02:36 PM   #3
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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.
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September 13th, 2010, 02:54 PM   #4
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Re: Correlation with Cards

E(XY)=1/52, E(X)E(Y) = (4/52)(13/52)=1/52.
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September 21st, 2010, 11:15 AM   #5
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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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