Question

When parking a car in a downtown parking lot, drivers pay according to the number of hours or fraction thereof. The probability distribution of the number of hours cars are parked has been estimated as follows:The cost of parking is 4.75 dollars per hour. Calculate the mean and standard deviation of the amount of revenue each car generates.

A. Mean =
B. Standard Deviation =

Answer 1

A. Mean = 4.24

B. Standard deviation = 2.79

A. Mean = $20.13

B. Standard deviation = $13.24

Explanation:

Given probability distribution:

The mean (μ) of a discrete probability function, often referred to as the expected value E(x), is calculated by multiplying each value of x by its corresponding probability and then summing these products.

`\text{E}(x)=\displaystyle \mu = \sum \left(x\cdot P(x)\right)`

Therefore, the mean of the given probability distribution is calculated as follows:

`\mu = 1\cdot 0.238 + 2\cdot 0.136 + 3\cdot 0.126+ 4\cdot 0.096 + 5\cdot 0.058 + 6\cdot 0.033 +7 \cdot 0.026 +8 \cdot 0.287`

`\mu = 0.238 + 0.272 + 0.378 + 0.384 + 0.29 + 0.198 +0.182 +2.296`

`\mu=4.238`

`\mu = 4.24\; \sf (2\;d.p.)`

So, the mean number of hours cars are parked is 4.24 hours (2 d.p.).

To calculate the standard deviation (σ), multiply the square of each value of x by its corresponding probability, sum these products, then subtract the square of the expected value (μ²), and finally square root the result.

`\displaystyle \sigma = √(\sum(x^2\cdot P(x))-\mu^2)`

Therefore, the standard deviation of the given probability distribution is calculated as follows:

`\sigma=√(25.732-4.238^2)`

`\sigma=√(25.732-17.960644)`

`\sigma=√(7.771356)`

`\sigma=2.787715193...`

`\sigma=2.79\; \sf (2\;d.p.)`

So, the standard deviation of the number of hours cars are parked is 2.79 hours (2 d.p.).

`\hrulefill`

Since the cost is $4.75 per hour, we can calculate the mean and standard deviation of the revenue by multiplying them by the cost.

`\begin{aligned}\textsf{Mean Revenue}&= \textsf{Mean $(\mu)$} * \textsf{Cost per hour}\\\\&=4.238 * \$4.75\\\\&=\$20.1305\\\\&=\$20.13\end{aligned}`

Please note that I have used the exact mean, not the rounded mean, to calculate the mean revenue. If we use the rounded mean of 4.24, then the mean revenue is $20.14.

`\begin{aligned}\textsf{Standard Deviation of Revenue}&= \textsf{Standard deviation $(\sigma)$} * \textsf{Cost per hour}\\\\&=2.787715193... * \$4.75\\\\&=\$13.241647...\\\\&=\$13.24\end{aligned}`

Please note that I have used the exact standard deviation, not the rounded standard deviation, to calculate the standard deviation of revenue. If we use the rounded standard deviation of 2.79, then the standard deviation of revenue is $13.25.