Consider n independent tosses of a die. Each toss has the probability pi of resulting in i. Let Xi be the number of tosses that result in i. Show...

Consider n independent tosses of a die. Each toss has the probability pi of resulting in i. Let Xi be the number of tosses that result in i. Show that X1 and X2 are negatively correlated (i.e. a large number of ones suggest a smaller number of two).

Since the total number of tosses is n, and tosses can either result in 1 or 2,we will have X1 + X2 = n ⇒ X1 = n − X2 . HenceCorr(X1 , X2 ) = Cov (X1 , X2 )Cov (X1 , n − X1 )V ar(X1 )=...