anyone would help me in a basic example?
a fair coin is tossed, n times. X is the number of Head and Y is the number of Tails. what is the COV(X,Y).
2026-03-29 14:54:34.1774796074
statistics basic question on covariance
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$$ X+Y = n $$ therefore $$ E(Y) = E(n) - E(X) = n - E(x) $$ now we have $$ \mathrm{Cov}(X,Y) = E\left[(X-E(X))(Y-E(Y))\right] $$
but we know that $$ Y-E(Y) = n-X - \left(n-E(x)\right)= -X + E(X) = -(X-E(X)) $$ thus $$ \mathrm{Cov}(X,Y) = E\left[(X-E(X))\cdot -(X-E(X))\right] = -E\left[(X-E(X))^2\right] = -\mathrm{Var}(X) = -np(1-p) $$ so you are right with $\mathrm{Cov}(X,Y) = -\dfrac{n}{4}$