Question

The waiting time, in hours, between successive speeders spotted by a radar unit is a continuous random variable with cumulative distribution function f(x) = 0, x< 0, 1 − e −8x, x≥ 0. find the probability of waiting less than 12 minutes between successive speeders (a) using the cumulative distribution function of x; (b) using the probability density function of x.

Answer 1

Given the CDF

$F_X(x)=\begin{cases}0&\text{for }x$

we have PDF

$\dfrac{\mathrm dF_X(x)}{\mathrm dx}=f_X(x)=\begin{cases}0&\text{for }x$

Note that $X$ is wait-time given in hours, so we need to convert from minutes to hours:

$12\text{ min}\times\dfrac{1\text{ hr}}{60\text{ min}}=\dfrac15\text{ hr}$

so we're looking for $\mathbb P\left(X$.

The CDF gives us this value right away, since $F_X(x)=\mathbb P(X$ for any continuous random variable $X$ with distribution function $F_X(x)$:

$\mathbb P\left(X$

To use the PDF, we need to integrate:

$\mathbb P\left(X$