The waiting time, $W$, of a traveler queuing at a taxi rank is distributed according to the cumulative distribution function, $G(w)$, defined by:
$$G(w) = \begin{cases}
0 & \text{ for } w<0,\\
1 – \left(\frac{2}{3}\right)e^\left(\frac{-w}{2}\right) & \text{ for } 0\le w < 2, \\
1 & \text{ for } w\ge 2
\end{cases}$$Is the random variable, $W$, discrete, continuous or mixed?
The solution provided was:
We see the distribution is mixed, with discrete 'atoms' at 0 and 2.
I don't understand the solution. Can I have more details please?
My answer was that the random variable, $W$ is continuous because it represents waiting time and time is a continuous variable. Why is my answer wrong?
Best Answer
It's wrong because - as the answer explained - there are discrete atoms at 0 and 2.
By that cdf, you can wait exactly 0 time with positive probability (similarly with 2). Because of that, the waiting time is mixed, not continuous.
Presumably you've been given definitions of all three. How are continuous r.v.s defined?
If it's not immediately clear from the formula, it often helps to draw the cdf: