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

  • 3
    $\begingroup$ What is the basis for saying "the covariance between two binomial variables is -np(1-p)? What are you assuming? $\endgroup$ – Glen_b -Reinstate Monica Jan 24 '16 at 8:58
  • $\begingroup$ As a special case of multinomial distribution. $\endgroup$ – Harry Jan 24 '16 at 12:46
  • 1
    $\begingroup$ 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? $\endgroup$ – 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 $$\text{Cov(X, Y)}=-Var(X)=-np(1-p)$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.