Question

The first two moments about origin
of a distribution are 1 and 21, then
the second moment about
mean=​

Answer 1

Answer:

By definition, the third central moment of a random variable X is E[(X−μ)3] where μ=E[X]

.

The Binomial(n,p

) random variable X can be written as the sum of n independent Bernoulli(p) random variables Yi, but we want to subtract the mean, so write X−np=∑nj=1Zj where

Zj=Yj−p={1−p−p with probability p with probability 1−p

are independent. Now

E[(X−np)3]=∑i=1n∑j=1n∑k=1nE[ZiZjZk]

There are n terms where i=j=k, and E[Z3i]=p(1−p)3−p3(1−p). The other terms are either of the form E[ZiZ2j] with i≠j or E[ZiZjZk] with i,j,k all different, and those are all 0 since the Z's are independent and have mean 0. So we conclude that the third central moment is

E[(X−np)3]=n(p(1−p)3−p3(1−p))

which can be simplified to np(1−p)(1−2p).

Explanation: