负二项分布
p{X=k} = f(k;r,p) = (k+r-1)!/[k!(r-1)!]p^r(1-p)^k, k=0,1,2,..., 0<p<1, r>0.
EX = sum(k=0->正无穷)kf(k;r,p) = sum(k=1->正无穷)k(k+r-1)!/[k!(r-1)!]p^r(1-p)^k = sum(k=1->正无穷)(k+r-1)!/[(k-1)!(r-1)!]p^r(1-p)^k
= r(1-p)/p*sum(k=1->正无穷)(k-1 + r+1 -1)!/[(k-1)!(r+1 -1)!]p^(r+1)(1-p)^(k-1)【把k-1看做1个整体,r+1看做1个整体,p和(1-p)的指数凑成(k-1)和(r+1)的形式】
= r(1-p)/p*sum(n=k-1=0->正无穷)(n+s-1)!/[n!(s-1)!]p^s(1-p)^n【n=k-1,s=r+1】
= r(1-p)/p*sum(n=0->正无穷)f(n;s,p)
= r(1-p)/p*1【由归一性,sum(n=0->正无穷)f(n;s,p)=1】
= r(1-p)/p
EX^2 = sum(k=0->正无穷)k^2f(k;r,p) = sum(k=1->正无穷)k^2(k+r-1)!/[k!(r-1)!]p^r(1-p)^k = sum(k=1->正无穷)k(k+r-1)!/[(k-1)!(r-1)!]p^r(1-p)^k
= sum(k=1->正无穷)(k-1+1)(k+r-1)!/[(k-1)!(r-1)!]p^r(1-p)^k
= sum(k=1->正无穷)(k-1)(k+r-1)!/[(k-1)!(r-1)!]p^r(1-p)^k
+ sum(k=1->正无穷)(k+r-1)!/[(k-1)!(r-1)!]p^r(1-p)^k
sum(k=1->正无穷)(k+r-1)!/[(k-1)!(r-1)!]p^r(1-p)^k
=EX= r(1-p)/p
sum(k=1->正无穷)(k-1)(k+r-1)!/[(k-1)!(r-1)!]p^r(1-p)^k
=sum(k=2->正无穷)(k-1)(k+r-1)!/[(k-1)!(r-1)!]p^r(1-p)^k
=sum(k=2->正无穷)(k+r-1)!/[(k-2)!(r-1)!]p^r(1-p)^k
=r(r+1)(1-p)^2/p^2sum(k=2->正无穷)(k-2 + r+2 -1)!/[(k-2)!(r+2 -1)!]p^(r+2)(1-p)^(k-2)
=r(r+1)(1-p)^2/p^2sum(n=k-2=0->正无穷)(n+s-1)!/[n!(s-1)!]p^s(1-p)^n 【n=k-2,s=r+2】
=r(r+1)(1-p)^2/p^2sum(n=0->正无穷)f(n;s,p)
=r(r+1)(1-p)^2/p^2,
EX^2 = sum(k=1->正无穷)(k-1)(k+r-1)!/[(k-1)!(r-1)!]p^r(1-p)^k
+ sum(k=1->正无穷)(k+r-1)!/[(k-1)!(r-1)!]p^r(1-p)^k
= r(r+1)(1-p)^2/p^2 + r(1-p)/p
DX = EX^2 - (EX)^2
= r(r+1)(1-p)^2/p^2 + r(1-p)/p - [r(1-p)/p]^2
= [r(r+1)(1-p)^2 + rp(1-p) - r^2(1-p)^2]/p^2
= r(1-p)[(r+1)(1-p) + p - r(1-p)]/p^2
= r(1-p)[1-p + p]/p^2
= r(1-p)/p^2
几何分布
p{X=k} = p(1-p)^(k-1), k=1,2,...,0<p<1.
EX = sum(k=1->正无穷)kp(1-p)^(k-1),
g(x) = sum(k=1->正无穷)x^k = 1/(1-x), 0 < x < 1.
g'(x) = sum(k=1->正无穷)kx^(k-1) = [1/(1-x)]' = 1/(1-x)^2,
EX = sum(k=1->正无穷)kp(1-p)^(k-1) = psum(k=1->正无穷)k(1-p)^(k-1)
= pg'(1-p) = p/[1-(1-p)]^2 = p/p^2 = 1/p,
EX^2 = sum(k=1->正无穷)k^2p(1-p)^(k-1) = sum(k=1->正无穷)k(k-1)p(1-p)^(k-1) + sum(k=1->正无穷)kp(1-p)^(k-1)
= sum(k=1->正无穷)k(k-1)p(1-p)^(k-1) + EX
g''(x) = sum(k=1->正无穷)k(k-1)x^(k-2) = [1/(1-x)^2]' = 2/(1-x)^3
EX^2 = sum(k=1->正无穷)k(k-1)p(1-p)^(k-1) + EX
= p(1-p)sum(k=1->正无穷)k(k-1)(1-p)^(k-2) + EX
= p(1-p)g''(1-p) + 1/p
= p(1-p)*2/[1-(1-p)]^3 + 1/p
= 2(1-p)/p^2 + 1/p
DX = EX^2 - [EX]^2 = 2(1-p)/p^2 + 1/p - (1/p)^2 = 1/p^2 - 1/p
= (1-p)/p^2
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