Question

一道概率题，麻烦给下过程，多谢。

Answer 1

EX=∑kP{X=k} =∑k(a^k)/[(1+a)^(k+1)]=(1/a)∑k(a/(1+a))^(k+1)
现来求无穷级数S(x)=∑k(x^(k+1)) |x|<1
S(x)=∑(k+2)(x^(k+1))-2∑x^(k+1)=[∑x^(k+2)]'-2x/(1-x)=(2x-x²)/(1-x)²-2x/(1-x)=x²/(1-x)²
EX=(1/a)∑k(a/(1+a))^(k+1)=(1/a)S(a/(1+a))=(1/a)[a²]=a
EX²=(1/a)∑k²(a/(1+a))^(k+1)
令G(x)=∑k²(x^(k+1)) ,G(x)=x²∑k²(x^(k-1))=x²[∑k(x^k)]'=x²[x∑k(x^(k-1))]'=x²[x[∑x^k]']'
=x²[x[1/(1-x)]']'=x²[1/(1-x)²]'=2x²/(1-x)³
EX²=(1/a)G(a/(1+a))=(1/a)*[2a²(1+a)]=2a(1+a)
DX=EX²-(EX)²=2a+a²

综上：EX=a, DX=2a+a²

Answer 2

题目?看不到