A toll road charges $ 4 for passenger cars and $ 7 for othervehicles. LetX1be the number of passenger cars and X2 be the number ofother vehicles entering the toll road in one hour. Assume thatX1andX2are independent normal random variables with means μ1= 45, μ2= 20, andstandard deviations σ1= 5, σ2= 6. a) Express the toll road revenue per hourY as a function of X1 and X2. b) Find the mean μY and the standard deviation σY of Y. c) Find the probability that the toll road revenue during a particular hour is between $300 and $400 inclusive.
A toll road charges $ 4 for passenger cars and $ 7 for othervehicles. LetX1be the number of passenger cars and X2 be the number ofother vehicles entering the toll road in one hour. Assume thatX1andX2are independent normal random variables with means μ1= 45, μ2= 20, andstandard deviations σ1= 5, σ2= 6.
a) Express the toll road revenue per hourY as a
b) Find the
c) Find the probability that the toll road revenue during a particular hour is between $300 and $400 inclusive.
Introduction:
It is given that X1 has a normal distribution with parameters, mean, μ1 = 45, standard deviations σ1 = 5; X2 has a normal distribution with parameters, mean, μ2 = 20, standard deviations σ2 = 6.
a.
Since X1 represents the number of passenger cars entering the toll road in one hour, and each passenger car is charged $4, the total amount charged for the passenger cars entering the toll road in one hour is $4X1.
Since X2 represents the number of other cars entering the toll road in one hour, and each other car is charged $7, the total amount charged for the other cars entering the toll road in one hour is $7X2.
The total toll road revenue per hour, Y is the total amount charged for all passenger and other cars entering the toll road in one hour, that is, $ (4X1 + 7X2).
Hence, the required expression is, Y = 4X1 + 7X2.
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