Question

概率论与数理统计，概率密度求系数

Answer 1

f(x) = Ae^(-|x|)

(1)

∫(-∞->+∞) f(x) dx =1

A∫(-∞->0) e^x dx +A∫(0->+∞) e^(-x)  dx =1

2A = 1

A=1/2

(2)

P(0<X<1)

=∫(0->1) f(x) dx 

=(1/2) ∫(0->1) e^(-x)  dx 

= -(1/2)[ e^(-x) ]|(0->1)

=(1/2)[ 1 - e^(-1) ] 

(3)

case 1: x≤0

F(X) 

= ∫(-∞->x) f(t) dt 

= (1/2)∫(-∞->x) e^t dt

= (1/2) e^x 

case 2: x>0

F(X) 

= ∫(-∞->x) f(t) dt 

=(1/2)∫(-∞->0) e^t dt +  (1/2)∫(0->x) e^(-t)  dt

= 1/2 + (1/2)[ 1- e^(-x) ] 

=1 - (1/2)e^(-x)