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 followsA. Mean=B. Standard Deviation=The cost of parking is 5 dollars per hour. Calculate the mean and standard deviation of the amount of revenue each car generates.A. Mean=B. Standard DeviationX 1 2 3 4 5 6 7 8P(X) 0.229 0.113 0.114 0.076 0.052 0.027 0.031 0.358

Answer 1

Answer:

u = 4.604 , s = 2.903

u' = 23.025 , s' = 6.49

Step-by-step explanation:

Solution:

- We will use the distribution to calculate mean and standard deviation of random variable X.

- Mean = u = E ( X ) = Sum ( X*p(x) )

 u = 1*0.229 + 2*0.113 + 3*0.114 + 4*0.076 + 5*0.052 + 6*0.027 + 7*0.031 + 8*0.358.

 u = 4.604

- Standard deviation s = sqrt ( Var ( X ) = sqrt ( E ( X^2) + [E(X)]^2

 s = sqrt [ 1*0.229 + 4*0.113 + 9*0.114 + 16*0.076 + 25*0.052 + 36*0.027 + 49*0.031 + 64*0.358 - 4.604^2 ]

s = sqrt ( 8.429184 )

s = 2.903          

- We will use properties of E ( X ) and Var ( X ) as follows:

- Mean = u' = E (Rate*X) = Rate*E(X) = $5*u =

               u' = $5*4.605

               u' = 23.025

- standard deviation = s' = sqrt (Var (Rate*X) ) = sqrt(Rate)*sqrt(Var(X)) = sqrt($5)*s =

               s' = sqrt($5)*2.903

               u' = 6.49

Answer 2

The mean and standard deviation for the amount of revenue each car generates include:

• u = 4.604 , s = 2.903

• u' = 23.025 , s' = 6.49

What is Mean?

This is defined as the average of a data set.

Distribution will be used to calculate mean and standard deviation

Mean = u = E ( X ) = Sum ( X*p(x) )

u = 1×0.229 + 2×0.113 + 3×0.114 + 4×0.076 + 5×0.052 + 6×0.027 + 7×0.031 + 8×0.358.

  = 4.604

Standard deviation (s) = sqrt ( Var ( X ) = sqrt ( E ( X²) + [E(X)]²

s = sqrt [ 1×0.229 + 4×0.113 + 9×0.114 + 16×0.076 + 25×0.052 + 36×0.027 + 49×0.031 + 64×0.358 - 4.604² ]

s = sqrt ( 8.429184 )

s = 2.903        

Properties of E ( X ) and Var ( X ) as follows:

Mean = u' = E (Rate×X) = Rate×E(X) = $5×u

u' = $5×4.605

u' = 23.025

Standard deviation (s' )= sqrt (Var (Rate×X) ) = sqrt(Rate)×sqrt(Var(X)) = sqrt($5)×s

= sqrt($5)*2.903

u' = 6.49.

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