Answer to Question #110906 in Statistics and Probability for Manoj

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

Exponential Distribution Probability 

$$F(x;\lambda) = \begin{cases} 0 &\text{if } x<0 \\ 1-e^{-\lambda x} &\text{if } x\geq0 \end{cases}$$

Given $\beta=200, \lambda=\dfrac{1}{\beta}=\dfrac{1}{200}$

 i) less than 200 months

$$P(X<200)=F(200;\lambda)=1-e^{-{1 \over 200}( 200)} =$$

$$=1-e^{-1}\approx0.63212$$

ii) between 100 months to 25 years.

$$P(100<X<3000)=F(3000;\lambda)-F(100;\lambda)=$$

$$=1-e^{-{1 \over 200}( 3000)} -(1-e^{-{1 \over 200}( 100)} )=$$

$$=e^{-{1 \over 2}}-e^{-15}\approx0.60653066-0.00000031\approx0.60653$$