Solved – Covariance in binomial distribution

Question

Is the covariance between number of success and failure in a binomial distribution with parameters n and p, the same as the covariance between two binomial variables, which is
-np(1-p)?

Answer 1

In a Binomial $\mathcal{B}(n,p)$ distribution, if $$X\sim\mathcal{B}(n,p)$$ is the number of successes, $$Y=n-X$$ is the number of failures. Therefore, $$\text{Corr}(X,Y)=-\text{Corr}(X,X)=-1$$

Then, since we know $$\text{Cov}(X,Y)=\text{Corr}(X, Y)\text{Stdev}(X)\text{Stdev}(Y)$$ and $$\text{Stdev}(X)=\text{Stdev}(Y)$$ we can calculate the covariance as $$\text{Cov(X, Y)}=-Var(X)=-np(1-p)$$