The useful life X of a particular kind of machine is a random variable with density function $$f ( x ) = \left\{ \begin{array} { c l }
\frac { 4 } { 15 } + \frac { 2 } { 3 x ^ { 2 } } & \text { if } 2 \leq x \leq 5 \
0 & \text { otherwise }
\end{array} \right.$$ where x is the number of years a randomly selected machine stays in use. $$P ( X \leq 3 ) = \frac { 28 } { 45 }$$

Question

The useful life X of a particular kind of machine is a random variable with density function $$f ( x ) = \left\{ \begin{array} { c l } 
\frac { 4 } { 15 } + \frac { 2 } { 3 x ^ { 2 } } & \text { if } 2 \leq x \leq 5 \
0 & \text { otherwise }
\end{array} \right.$$ where x is the number of years a randomly selected machine stays in use. $$P ( X \leq 3 ) = \frac { 28 } { 45 }$$

Quizplus · Accepted Answer

The probability density function is valid because it satisfies the following conditions:1. $f(x) \geq 0$ for all $x$.2. $\int_{-\infty}^{\infty} f(x) dx = 1$.The probability that the machine stays in use for less than or equal to 3 years is given by:$$P(X \leq 3) = \int_{-\infty}^{3} f(x) dx = \int_{2}^{3} \left(\frac{4}{15} + \frac{2}{3x^2}\right) dx = \frac{28}{45}$$Therefore, the given probability is correct.

The Useful Life X of a Particular Kind of Machine