Question:
Consider an amusement park where visitors are arriving according to a Poisson process with rate 1. Upon arrival, a visitor spends a random amount of time in the park and then departs. The time spent by the visitors are independent of one another, as well as of the arrival process, and have common probability density function
\[
f(x) = \begin{cases}
e^{-x}, & x>0,
0, & \text{otherwise}.
\end{cases}
\]
If at a given time point, there are 10 visitors in the park and \( p \) is the probability that there will be exactly two more visitors before the next departure, then
\[
p = _________ \text{ (round off to 2 decimal places).}
\]
Consider an amusement park where visitors are arriving according to a Poisson process with rate 1. Upon arrival, a visitor spends a random amount of time in the park and then departs. The time spent by the visitors are independent of one another, as well as of the arrival process, and have common probability density function
\[
f(x) = \begin{cases}
e^{-x}, & x>0,
0, & \text{otherwise}. \end{cases} \] If at a given time point, there are 10 visitors in the park and \( p \) is the probability that there will be exactly two more visitors before the next departure, then \[ p = _________ \text{ (round off to 2 decimal places).} \]
0, & \text{otherwise}. \end{cases} \] If at a given time point, there are 10 visitors in the park and \( p \) is the probability that there will be exactly two more visitors before the next departure, then \[ p = _________ \text{ (round off to 2 decimal places).} \]
Show Hint
For Poisson processes, use the properties of the Poisson distribution to compute probabilities for events occurring within a given time frame.
Updated On: Dec 29, 2025
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Correct Answer: 143
Solution and Explanation
The probability of exactly two more visitors before the next departure is modeled using the Poisson distribution. The probability \( p \) is given by:
\[
p = e^{-1} \frac{1^2}{2!} \approx 0.50.
\]
Thus, the value is \( 0.50 \).
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