MeanOfBinomialDistribution

Mean of binomial distribution (English)

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If X ∼ B(n,p), i.e. X is distributed binomially with parameters n and p, then:

〈X 〉 =

n
∑
r=0

rp_r =

n
∑
r=0




n
r




p ^r (1− p ) ^n−r

n
∑
r=1

(r−1)!(n−r)!

p^r (1−p)^n−r

= np

n
∑
r=1

(n−1)!

(r−1!)(n−r)!

p^r−1 (1−p) ^n−r

= np

n−1
∑
r=0




n−1
r




p^r (1−p)^n−1−r

= np(p+(1−p))ⁿ⁻¹

= np