Mathematics: Arandom variable x is exponentially distributed with a mean of 25. what is the probability that x is between 15 and 35?

Arandom variable x is exponentially distributed with

Ответ:

holmesleauja

02.01.2022

The probability density function for

is

$f_X(x)=\begin{cases}\dfrac1\beta e^{-x/\beta}&\text{for }x0\\\\0&\text{otherwise}\end{cases}$

where $\beta$ is the scale parameter of the distribution, which for exponential distributions is also the mean.

The probability is then

$\mathbb P(15$
$=\displaystyle\frac1{25}\int_{15}^{35}e^{-x/25}\,\mathrm dx$
$=\displaystyle-e^{-x/25}\bigg|_{x=15}^{x=35}$
$=e^{-15/25}-e^{-35/25}\approx0.3022$