Mathematics, 06.08.2019 19:30 Violinartist7267
The number of people in a car that crosses a certain bridge is represented by the random variable x, which has a mean value μx = 2.7, and a variance σ2x = 1.2. the toll on the bridge is $3.00 per car plus $ .50 per person in the car. determine the mean and variance of the total amount of money that is collected from a car that crosses the bridge.
Answers: 1
be the total amount of money paid by any given set of passengers. If there are 
passengers in a car, then the driver must pay a toll of 
.![E[Y]=E[0.5X+3]=0.5E[X]+3E[1]=0.5\mu_X+3=\boxed{4.35}](/tpl/images/0172/2983/dd36b.png)
![E[Y^2]=E[0.25X^2+3X+9]=0.25E[X^2]+3E[X]+9E[1]=0.25E[X^2]+3\mu_X+9](/tpl/images/0172/2983/9dde1.png)
![\mathrm{Var}[X]=E[X^2]-E[X]^2\implies E[X^2]={\sigma^2}_X+{\mu_X}^2](/tpl/images/0172/2983/28b52.png)
![\implies E[Y^2]=0.25({\sigma^2}_X+{\mu_X}^2)+3\mu_X+9\approx19.22](/tpl/images/0172/2983/48952.png)
![\implies\mathrm{Var}[Y]=E[Y^2]-E[Y]^2=\boxed{0.3}](/tpl/images/0172/2983/2101d.png)
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