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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)?

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

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