Question

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.

Answer 1

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

Then $Y$ has first moment (equal to the mean)

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

and second moment

$E[Y^2]=E[0.25X^2+3X+9]=0.25E[X^2]+3E[X]+9E[1]=0.25E[X^2]+3\mu_X+9$

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

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

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

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