Question

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

Answer 1

The probability density function for
`X` is


`f_X(x)=\begin{cases}\frac1\beta e^(-x/\beta)&\text{for }x>0\\\\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<X<35)=\displaystyle\int_(15)^(35)f_X(x)\,\mathrm dx`

`=\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`