>>766
そこから定義通りに
E[X] = Σ_k k*P(X=k)
E[X^2] = Σ_k k^2 *P(X=k)
を計算して
V[X] = E[X^2] - E[X]^2
が出る。
ベルヌイなら
E[X] = 0*P(X=0) + 1*P(X=1) = p
E[X^2] = 0*P(X=0) + 1*P(X=1) = p
V[X] = p-p^2 = p(1-p) = pq

2項分布は
E[X] = 0*P(X=0) + 1*P(X=1) + 2*P(X=2) + …
= Σ_{k=1 to n} k (nCk) p^k q^(n-k)
= Σ_{k=1 to n} n ((n-1)C(k-1)) p^k q^(n-k)
= np Σ_{k=1 to n} ((n-1)C(k-1)) p^(k-1) q^(n-k)
= np (p+q)^(n-1) = np

E[X^2] = Σ_{k=1 to n} k^2 (nCk) p^k q^(n-k)
= np Σ_{k=1 to n} k ((n-1)C(k-1)) p^(k-1) q^(n-k)
= np { Σ_{k=2 to n} (n-1) ((n-2)C(k-2)) p^(k-1) q^(n-k)
Σ_{k=1 to n} ((n-1)C(k-1)) p^(k-1) q^(n-k)}
= np { (n-1)p + 1} =np (np +q) = (np)^2 + npq
V[X] = E[X^2] - E[X]^2 = npq