If $X\sim \operatorname{Bin}(N,p)$, then what is the distribution of $Y = N-X$?

Question

Let $X\sim\text{Bin}(N,p).$ What is the distribution of $Y= N- X$?

Is it $Y\sim \text{Bin}(N,1-p)$?

Also, what is the mixed moment of $E(XY)$? What is the $\text{Cov}(X,Y)$?

Many thanks here.

Answer 1

Yes.   If $X$ is the count of successes in $N$ Bernoulli trials, then $N-X$ is the count of the failures.   This will have a distribution of: $\mathcal{Bin}(N, 1-p)$.

Since $Y=N-X$ then the Linearity of Expectation will tell you:

$$\mathsf E(XY)=N~\mathsf E(X)-\mathsf E(X^2)$$

You can evaluate this knowing that $X$ is Binomially distributed.

Similarly Bilinearity of Covariance says:

$$\mathsf{Cov}(X,Y)= \mathsf {Cov}(X,N)-\mathsf {Cov}(X,X)$$