The expectation and variance of 0 1 distribution are expected p variance p( 1-p), binomial distribution expected np, variance np( 1-p).

Probability distribution is very important in mathematics, physics and engineering, and has great influence in many aspects of statistics. If the random variable X obeys a Gaussian distribution with a mathematical expectation of μ and a variance of σ 2, it is recorded as N(μ, σ 2).

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For fixed n and p, when k increases, the probability P{X=k} first increases until it reaches the maximum value, and then monotonically decreases. It can be proved that the general binomial distribution also has this property, and when (n+ 1)p is not an integer, when k=[(n+ 1)p], the binomial probability P{X=k} reaches the maximum.

When (n+ 1)p is an integer, the binomial probability P{X=k} reaches the maximum when k=(n+ 1)p and k=(n+ 1)p- 1. [x] is an integer function, that is, the largest integer that does not exceed X.