Question

The TV repair company receives TVs with various faults. Therefore, their correction time Tis a random variable. Find the probability density function of this random dudge if the probability distribution function F(t) has the following form: (0, t <0, F(t) ={1-, t>0,

Answer 1

$$F(t) = \begin{cases} 0, &t<0 \\ 1-e^{-t}, &t\geq0 \end{cases}$$

$$f(t)=\dfrac{dF}{dt}$$

$$f(t) = \begin{cases} 0, &t<0 \\ e^{-t}, &t\geq0 \end{cases}$$

$$\displaystyle\int_{-\infin}^{\infin}f(t)dt=\displaystyle\int_{0}^{\infin}e^{-t}dt$$

$$=\lim\limits_{A\to \infin}[-e^{-t}]\begin{matrix} A \\ 0 \end{matrix}=-0+1=1$$

$f(t) = \begin{cases} 0, &t<0 \\ e^{-t}, &t\geq0 \end{cases}$