Property 0: B(n, p) is a valid probability distribution
Proof: the main thing that needs to be proven is that
where f(x) is the pdf of B(n, p). This follows from the well-known Binomial Theorem since
The Binomial Theorem that
can be proven by induction on n.
Proof (mean): First we observe
where m = n − 1 and i = k − 1 . But
where fm,p(i) is the pdf for B(m, p), and so we conclude μ = E[x] = np.
We begin using the same approach as in the proof of the mean: