Question

Calculate P(X < 8) if:
(a) X~Unif[5;10] (b) X~Exp(0:5)
(c) X~N(10;4) (d) X~LN(2;4)

Answer 1

a)

X~Uni(5,10). So, $f(x)=\frac{1}{b-a}=\frac{1}{10-5}=0.2,5\leq x\leq 10$

$P(X<8)=\int_{5}^{8}f(x)dx=\int_{5}^{8}0.2dx=0.2[x]_{5}^{8}=0.2[8-5]=0.6$

b)

X~exp(0.5). So, $F(x)=P(X<x)=1-e^{-\lambda x}=1-e^{-0.5 x}$

$P(X<8)=F(8)=1-e^{-0.5(8)}=0.9817$

c)

X~N(10,4)

$P(X<8)=P(Z<\frac{8-10}{\sqrt{4}})=P(Z<-1)=P(Z>1)=1-P(Z<1)=1-0.8413=0.1587$

d)

X~LN(2,4)

So, Ln(X)~N(2,4)

$P(X<8)=P(Ln(X)<Ln(8))=P(Z<\frac{Ln(8)-2}{\sqrt{4}})=P(Z<0.04)=0.5160$