二项分布方差公式推导

二项分布方差公式推导

若ξ～B(n,p)，q=1-p ，求证D ξ=npq

∵E ξ=np ， kC n k p k q n-k =n p 11

k n C --p k-1q n-k , kk C n k p k q n-k =np[(k-1)11

k n C --p k-1q n-k ＋11k n C --p k-1q n-k ] =np[(n －1)p 22k

n C --p k-2q n-k ＋11k n C --p k-1q n-k

] 而D ξ=22()E E ξξ-，

∴D ξ=(1×1×C n 1p 1q n-1＋2×2 C n 2p 2q n-2＋…＋k ×k C n k p k q

n-k ＋…＋n ×n C n n p n q 0)2()

np - =np(1×C n-10p 0q n-1＋2C n-11p 1q n-2＋3C n-12p 2q n-2＋…＋

k C n-1k-1p k-1q n-k

＋…＋n C n-1n-1p n-1q 0)－2np E ξ＋n 2p 2(p ＋q)n

=np{[0×C n-10p 0q n-1＋1C n-11p 1q n-2＋2C n-12p 2q n-2＋…＋

(k-1) C n-1k-1p k-1q n-k ＋…＋(n-1)C n-1n-1p n-1q 0]＋(C n-10p 0q n-1＋

C n-11p 1q n-2＋C n-12p 2q n-2＋…＋C n-1k-1p k-1q n-k ＋…＋

C n-1n-1p n-1q 0)}2()np -

=np[E η＋(p ＋q)n-1] 2()

np - =np[(n －1)p ＋1] 2()

np -

=np(1－p)

=npq .