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Dilip Sarwate
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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) = \left\{ \begin{align} 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{align} \right.$$$G(w) = \begin{cases} 0 & \text{ for } w<0,\\ 1 - \left(\frac{2}{3}\right)e^\left(\frac{-w}{2}\right) & \text{ for } 0\le w < 2, \\ 1 & \text{ for } w\ge 2 \end{cases}$$

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?

The waiting time, $W$, of a traveler queuing at a taxi rank is distributed according to the cumulative distribution function, $G(w)$, defined by:

$G(w) = \left\{ \begin{align} 0 && \text{ for } w<0 &\\ 1 - \left(\frac{2}{3}\right)e^\left(\frac{-w}{2}\right) && \text{ for } 0\le w < 2 &\\ 1 && \text{ for } w\ge 2 \end{align} \right.$

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?

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?

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mauna
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Why is this random variable both continuous and discrete?

The waiting time, $W$, of a traveler queuing at a taxi rank is distributed according to the cumulative distribution function, $G(w)$, defined by:

$G(w) = \left\{ \begin{align} 0 && \text{ for } w<0 &\\ 1 - \left(\frac{2}{3}\right)e^\left(\frac{-w}{2}\right) && \text{ for } 0\le w < 2 &\\ 1 && \text{ for } w\ge 2 \end{align} \right.$

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?