Question

The life of a compressor manufactured by a company is known to be 200 months on an
average following an exponential distribution .Find the probability that the life of a
compressor is i) less than 200 months ii) between 100 months to 25 years.

Answer 1

let X=The life of a compressor manufactured by a company 

$\mu=200months\\ \lambda=\frac{1}{\mu}=\frac{1}{200}=0.005$

here $X\thicksim exp(0.005)$

$f(x)= \begin{cases} 0.005e^{(-0.005x)} &\text{if } x\ge0 \\ 0 &\text{if } x<0 \end{cases}$

and

$F(x)= \begin{cases} 1-e^{(-0.005x)} &\text{if } x\ge0 \\ 0 &\text{if } x<0 \end{cases}$

i)

$P(X<200)=F(200)\\ P(X<200)=1-e^{(-0.005*200)}\\ P(X<200)=0.6321$

probability that the life of a compressor is less than 200 months=0.6321

ii)

$P(100<X<300)=F(300)-F(100)\\ P(100<X<300)=(1-e^{(-0.005*300)})\\ \hspace{12 em}-(1-e^{(-0.005*100)})\\ P(100<X<300)=0.3834$

probability that the life of a compressor is between 100 months to 25 years=0.3834