Respuesta :
Answer:
The probability that a repair time exceeds 8 hours is [tex]P(X>8)\approx 0.000746586[/tex]
The conditional probability that a repair takes at least 13 hours, given that it takes more than 9 hours is [tex]P(X\geq 13 | X>9)\approx 0.0273237376[/tex]
Step-by-step explanation:
A continuous random variable X is said to have an exponential distribution with parameter [tex]\lambda > 0[/tex] shown as [tex]X \sim Exponential(\lambda)[/tex], if its probability density function is given by
[tex]f_X(x)=\begin{cases}\lambda e^{-\lambda x} & x > 0\\ 0 &\text{otherwise}\end{cases}[/tex]
Let X denote the time require to repair a machine.
(a) The probability that a repair time exceeds 8 hours;
[tex]P(X>8)=1-P(X\leq 8)\\P(X>8)=1-\int\limits^8_0 {0.9e^{-0.9x}} \, dx \\P(X>8)=1-0.999253\\P(X>8)\approx 0.000746586[/tex]
(b) The conditional probability that a repair takes at least 13 hours, given that it takes more than 9 hours;
We want to find [tex]P(X\geq 13 | X>9)[/tex].
[tex]P(X\geq 13 | X>9)=\frac{P(X\geq 13)}{P(X>9)} \\\\P(X\geq 13 | X>9)=\frac{\int\limits^{\infty}_{13} {0.9e^{-0.9x}} \, dx}{\int\limits^{\infty}_{9} {0.9e^{-0.9x}} \, dx}\\ \\P(X\geq 13 | X>9)=\frac{8.29382\times10^{-6}}{0.000303539} \\\\P(X\geq 13 | X>9)\approx 0.0273237376[/tex]
