## Poisson distrubution / exponential random variable

An employee at a gym recorded the number of people coming in to work out or attend physical therapy during each half hour that he worked.

A. Explain how a Poisson distribution and an exponential distribution could be involved in his recording of data. Be very specific in how each distribution could apply.
B. His recorded data is: 3 5 2 3 4 3 6 4 3 0 1 2 3 2 0 6 2 4 2 3 4 2 4 1 5 5 0 2 3 5
Construct a frequency table comparing the data to an appropriate Poisson distribution. Does the comparison support the conjecture that the mean number of arrivals is approximately Poisson?

# of people Frequency
0 3
1 2
2 7
3 7
4 5
5 4
6 2
30
C. The time between people arriving is:
Time Frequency
1 7
2 6
3 7
4 18
5 13
6 13
7 7
8 9
9 6
10 4
90
If X is an exponential random variable with mean $\theta=5$, compare the probability P(X>2) with the proportion of times that are greater than 2 from the data provided.