Question

Joe has 7 shirts and he ranks them: he has a favorite one, a second favorite etc. Each day of the week (=7 days) he wears a new one and the one worn that day goes to the laundry and stays in the laundry for the remainder of the procedure.The new one is picked randomly from those which are clean at that moment.

The following procedure starts on Monday: on any given day, if Joe wears his favorite shirt, he feels cool and drinks 7 glasses of beer that day in the pub, if he wears his second favorite shirt, he drinks 6 glasses that day, etc.

What is the covariance between the number of drinks on Monday and Wednesday?

Given Answer -2/3

Answer 1

Let X be the number or drinks on Monday and Y be the number of drinks on Wednesday, which is basically the rank of the T-shirt he likes considering we rank the most favourite as 7 and so on.

We have to find Cov(X,Y) =E(XY) - E(X)E(Y)

So we first find the joint pdf of X and Y.
Consider the probability that X = j and Y = k for 1<=j,k<=7 and j not equals k. Then since the T-shirt's are randomly distributed we can think of this as permutations of 7 numbered T-shirt's over 7 days. So the total number of ways of doing that would be 7! While the required number of ways for the probability required would be if we fix j for Monday and T-shirt k for Wednesday and permutate the rest in 5! Ways because 5 days are left apart from Monday and Wednesday.

So P(X = j, Y = k) = 5!/7! = 1/42 for j not same as k

So

which is basically product of all possible numbers from 1 to 7 with each other except for the squares divided by 42

Hence it is (1+2+3+4+5+6+7)2 - (sum of squares from 1 to 7)

Hence by the formula of sum of n natural numbers = (1/2)n(n+1) and sum of n squares = (1/6)n(n+1)(2n+1)

we get,

E(XY) = (1/42) [((7x8)/2)2 - (1/6)(7x8x15)] = (644/42)

And for each of X and Y marginal distributions we have the probability = 1/7 by the same way as we fix the required for the particular day and permute the rest in 6! Ways divided by 7! Ways of all possible permutations

So E(X) = E(Y) as they have same probabilities and can separately take all values from 1 to 7

= (1/7)(28) = 4 by applying the same formula for sum of n natural numbers

So Cov(X,Y) = 644/42 - 42 = -2/3

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