已知ξ的分布律为pi=P{ξ=i}=1/3*(2/3)^i,求ξ的数学期望

Question

已知ξ的分布律为pi=P{ξ=i}=1/3*(2/3)^i,求ξ的数学期望

Answer 1

E(ξ) = Σ(0到∞) i* 1/3*(2/3)^i
= 1/3 * Σ i*(2/3)^i
现在来计算级数 g(x) = Σ i*x^i,注意
Σ i*x^i = x Σ i*x^(i-1)
=x Σ (x^i)'
=x (Σ x^i)'
=x*(1/(1-x))'
=x/(1-x)²
于是 E(ξ) = 1/3 * g(2/3) = 2