Question2: 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 is given below
X 1 2 3 4 5 6 7 8
f(X) 0.30 0.24 0.17 0.10 0.08 0.05 0.04 0.02
a. Calculate the mean and standard deviation of the number of hours cars are parked in the lot.
b. The cost of parking cars $2.5 per hour. Calculate the mean and standard deviation of the amount of the revenue each car generates.
To calculate the mean and standard deviation of the number of hours cars are parked in the lot, you can use the following formulas:
Mean (or expected value) = Σ (X * f(X))
Standard deviation = √(Σ ((X - mean)^2 * f(X)))
Now let's calculate each part separately:
a. Mean and Standard Deviation of the number of hours cars are parked:
First, calculate the mean:
Mean = 1 * 0.30 + 2 * 0.24 + 3 * 0.17 + 4 * 0.10 + 5 * 0.08 + 6 * 0.05 + 7 * 0.04 + 8 * 0.02
Mean = 0.30 + 0.48 + 0.51 + 0.40 + 0.40 + 0.30 + 0.28 + 0.16
Mean = 2.83
The mean or average number of hours cars are parked in the lot is 2.83.
Next, calculate the standard deviation:
Standard deviation = √((1- mean)^2 * 0.30 + (2- mean)^2 * 0.24 + (3- mean)^2 * 0.17 + (4- mean)^2 * 0.10 + (5- mean)^2 * 0.08 + (6- mean)^2 * 0.05 + (7- mean)^2 * 0.04 + (8- mean)^2 * 0.02)
Standard deviation = √((1-2.83)^2 * 0.30 + (2-2.83)^2 * 0.24 + (3-2.83)^2 * 0.17 + (4-2.83)^2 * 0.10 + (5-2.83)^2 * 0.08 + (6-2.83)^2 * 0.05 + (7-2.83)^2 * 0.04 + (8-2.83)^2 * 0.02)
Standard deviation = √((0.8589) * 0.30 + (0.6561) * 0.24 + (0.1129) * 0.17 + (0.1072) * 0.10 + (0.2862) * 0.08 + (0.4624) * 0.05 + (0.5476) * 0.04 + (4.6464) * 0.02)
Standard deviation = √(0.2577 + 0.1571 + 0.0192 + 0.0107 + 0.0229 + 0.0231 + 0.0219 + 0.0929)
Standard deviation = √0.6055
Standard deviation ≈ 0.78
The standard deviation of the number of hours cars are parked in the lot is approximately 0.78.
b. Mean and Standard Deviation of the amount of revenue each car generates:
Since the cost of parking is $2.5 per hour, the mean and standard deviation of the revenue generated by each car can be calculated by multiplying the mean and standard deviation of the number of hours by $2.5:
Mean revenue = Mean * $2.5
Mean revenue = 2.83 * $2.5
Mean revenue = $7.075
The mean revenue generated by each car is $7.075.
Next, calculate the standard deviation of the revenue:
Standard deviation revenue = Standard deviation * $2.5
Standard deviation revenue = 0.78 * $2.5
Standard deviation revenue = $1.95
The standard deviation of the revenue generated by each car is $1.95.