Question

if the distribution of a car’s transmission is governed by exponential density function with mean 100000, what will be the probability that car’s transmission will fail during its first 50000 km of operation.

Answer 1

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

$\lambda=\dfrac{1}{100000}=10^{-5}$

$$P(X\leq50000)=1-e^{-10^{-5}(50000)}=1-e^{-0.5}\approx0.39347$$

$$\approx0.39347$$