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:

X 1 2 3 4 5 6 7 8
P(X) 0.238 0.136 0.126 0.096 0.058 0.033 0.026 0.287
A. Mean = ?
B. Standard Deviation = ?
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

μ = Σ (X * P(X))

σ = √[Σ ((X - μ)² * P(X))]

Mean (Expected Value):

μ = (1 * 0.238) + (2 * 0.136) + (3 * 0.126) + (4 * 0.096) + (5 * 0.058) + (6 * 0.033) + (7 * 0.026) + (8 * 0.287)

μ = 0.238 + 0.272 + 0.378 + 0.384 + 0.29 + 0.198 + 0.182 + 2.296

μ = 4.034 hours

Standard Deviation:

σ = √[(1-4.034)^2 * 0.238 + (2-4.034)² * 0.136 + (3-4.034)² * 0.126 + (4-4.034)² * 0.096 + (5-4.034)² * 0.058 + (6-4.034)² * 0.033 + (7-4.034)² * 0.026 + (8-4.034)² * 0.287]

σ = √[(9.66 * 0.238) + (4.139 * 0.136) + (1.0716 * 0.126) + (0.001224 * 0.096) + (0.092* 0.058) + (7.949 * 0.033) + (9.236 * 0.026) + (15.77 * 0.287)]

σ = √[2.301 + 0.563 + 0.134 + 0.000117 + 0.00535 + 0.2617 + 0.0602 + 4.532]

σ = √(7.858)

σ ≈ 2.80 hours

Therefore, mean is approximately 4.03 hours, and the standard deviation is approximately 2.80 hours.

Multiply both with $4.75 to get the mean and standard deviation of the amount of revenue each car generates.

μ = 4.03 * 4.75

μ = $19.14

σ = 2.80 * 4.75

σ = $13.30