統計離散均勻分配的公式推導

Mx(t)=E(exp(xt)=(1/k)[ ext(t)+ext(2t)+...+ext(kt)] -----(A)
=(1/k)exp(t)*[exp(kt)-1]/[exp(t)-1]
E(x)=d/dt Mx(t) (t=0) =Mx'(0)
(by (A)) = (1/k)[1+2+...+k]= (k+1)/2
Var(x)=E(x^2)-[E(x)]^2
=Mx"(0)- (k+1)^2/4
(by(A))=(1/k)[1^2+2^2+...+k^2] - [(k+1)/2]^2
=(k+1)(2k+1)/6 - (k+1)^2/4
=(k+1)(k-1)/12