pattern recognition and machine learning 2-4, 이항 분포 평균, 분산 증명

이항 분포의 평균이 

다음 식으로 주어진다는 것을 증명, (이항 분포의 평균은 횟수 * 그 확률)

We note that m = 0 actyakky doesn't affect the Expectation, so we let the summation begin from m = 1,

Moreover, in the second last step, we rewrite the subindex of the summation, and what we actually do is let,

k = m - 1, 

And in the last step, we have taken advantage of  binomial theorem, 

 

Variance is straightforward once Expectation has been calculated.

Here we will use a little tick, ,

-mu = -1 + (1 - mu)