Question

概率论，求D(X)

Answer 1

EX=[K*P*(1-p)^(K-1)]求和=[K*(1-P)^(K-1)]求和*p……P是常数，提到求和符号以外
= - [(1-P)^K]求导再求和*p
= - [(1-P)^K]求和再求导 * P 求和求导交换顺序
= -[(1-P)/P] 再求导 * p 计算求和项
= -[1/P^2] *P 求导
= 1/p

同理算 E（X平方）
E(X^2)+EX= [ (K+1)*K*P*(1-P)^(K-1) ] 求和
= [ (K+1)*K*(1-P)^(K-1) ] 求和*p ……常数p提到求和项以外
= [ (1-P)^(K+1) ]求二阶导 再求和*p
= [ (1-P)^(K+1) ]求和， 再求二阶导，再*p
= [ (1-P)^2 / P] 求二阶导，再*P
= [ 1/P-2+P] 求二阶导，再 *P
= [ 2*P^(-3) ]*P
=2/P^2
方差=E（x平方）-（EX)的平方=[E(X^2)+EX ]-EX-(EX)^2
=2/P^2-1/P-(1/P)^2
=1/P^2-1/P=(1-P)/P