# Covariance in binomial distribution

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)?

• What is the basis for saying "the covariance between two binomial variables is -np(1-p)? What are you assuming? – Glen_b -Reinstate Monica Jan 24 '16 at 8:58
• As a special case of multinomial distribution. – Harry Jan 24 '16 at 12:46
• Then I don't see any distinction between the two things you're asking about - aren't both halves of the question asking exactly the same things? – Glen_b -Reinstate Monica Jan 25 '16 at 0:17

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 $$\text{Cov(X, Y)}=-Var(X)=-np(1-p)$$