Answer to Question #259300 in Statistics and Probability for Saeed

Question

The lifetime T (years) of an electronic component is a continuous random variable with a

probability density function given by

( ) , 0 t

f t e t 

 

(a) Find the lifetime L which a typical component is 60% certain to exceed.

(b) If five components are sold to a manufacturer, find the probability that at least one of

them will have a lifetime less than L years

Answer 1

The lifetime $T$ (years) of an electronic component is a continuous random variable with a probability density function given by

$$f(t)=e^{-t}, t\geq0$$

(a)

$$P(T>L)=\displaystyle\int_{L}^{\infin}e^{-t}dt=\lim\limits_{A\to \infin}\displaystyle\int_{L}^{A}e^{-t}dt$$

$$=\lim\limits_{A\to \infin}\displaystyle[-e^{-t}]\begin{matrix} A \\ L \end{matrix}=e^{-L}$$

Given $P(T>L)=0.6.$ Then

$$e^{-L}=0.6$$

$$L=-\ln0.6$$

$$L\approx0.511\ years.$$

(b) Assuming that the lifetime of each component is independent we have

$$P(at\ least\ one\ component\ has \ T<0.511)$$

$$=1-P(no\ component\ has \ T<0.511)$$

$$=1-(0.6)^5=0.92224$$