Question

概率论与数理统计如图，题2怎么做？

Answer 1

f1(x)=(1/√2πσ)exp[-x^2/(2σ^2)]

x>0，f2(x)=λexp(-λx)

其他，f2(x)=0

f(x)=af1(x)+bf2(x)

x>0，f(x)=a*(1/√2πσ)exp[-x^2/(2σ^2)]+b*λexp(-λx)

其他，f(x)=a*(1/√2πσ)exp[-x^2/(2σ^2)]

F(x)=∫(-∞->x)f(x)dx

则F(0)=∫(-∞->0)f(x)dx

=∫(-∞->0)a*(1/√2πσ)exp[-x^2/(2σ^2)]dx=∫(-∞->0)a*(1/√2πσ^2)exp[-(x/σ)^2/2]d(x/σ)

=(a/σ^2)Φ(0)

=(a/σ^2)*1/2=1/8

得到a/σ^2=1/4

Φ(0)=1/2