What is covariance and correlation between two random variables regarding coin flips?

Question

So say Coin A has probability 1/3 of heads and Coin B has probability 2/3 of heads. You pick one of the two coins at random (50% chance of A or B) then flip that coin twice. If X is 0 if tails on first flip, 1 if heads on first flip, and Y is 0 if tails on second flip, 1 if heads on second flip, what would the covariance and correlation be for X and Y?

Looking for cov(X,Y) and cor(X,Y). I keep getting confused on how to set it up, and my numbers don't make sense.

Answer 1

Hi Zach E.,

There may be more elegant ways of calculating, but what you need here is a quick way of finding cov (first), then variances, and finally cor. Then you will look back ....

For cov, imagine a dataset exactly distributed as per the probabilties. Let's consider coin A, just:

p(X)=(0,1) = (2/3, 1/3)

and p(Y) = (0,1) = (2/3 , 1/3)

All right, a representative proportional dataset for 9 flips for (X,Y) would be: (0,0) x 4 , (0,1) x 2 , (1,0) x 2, and (1,1) x 1 .

Graph those data up and apply the SIGMA [(x-x(ave))*(y-y(ave)] / N formula, where x(ave)=(1/3), y(ave)=(1/3), N=9.

Unless I'm missing something, you end up with a zero numerator, there. --> ZERO covariance! So correlation will also be zero.

So -- look back: what did you just learn about random variables (X and Y)? THEY DON'T COVARY! THEY ARE INDEPENDENT! Which, you should have realized in advance?

The reason I had you calculate the covariance is, you WILL need to apply it when your variables are not independent. It's good to have some experience!

--Cheers, --Mr. d.