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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.

Respuesta :

LammettHash

Let [tex]Y[/tex] be the total amount of money paid by any given set of passengers. If there are [tex]X[/tex] passengers in a car, then the driver must pay a toll of [tex]Y=0.5X+3[/tex].

Then [tex]Y[/tex] has first moment (equal to the mean)

[tex]E[Y]=E[0.5X+3]=0.5E[X]+3E[1]=0.5\mu_X+3=\boxed{4.35}[/tex]

and second moment

[tex]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[/tex]

Recall that the variance is the difference between the first two moments:

[tex]\mathrm{Var}[X]=E[X^2]-E[X]^2\implies E[X^2]={\sigma^2}_X+{\mu_X}^2[/tex]

[tex]\implies E[Y^2]=0.25({\sigma^2}_X+{\mu_X}^2)+3\mu_X+9\approx19.22[/tex]

[tex]\implies\mathrm{Var}[Y]=E[Y^2]-E[Y]^2=\boxed{0.3}[/tex]

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