The number of people in a car that crosses a certain bridge is represented by the random variable X, which has

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. The mean and variance of the total amount of money that is collected from a car that crosses the bridge are:








(a) mean = $1.35, variance = $ .30.





(b) mean = $8.60, variance = $ .30.





(c) mean = $8.60, variance = $ .60.





(d) mean = $4.35, variance = $3.30.





(e) mean = $4.35, variance = $ .30.

The number of people in a car that crosses a certain bridge is represented by the random variable X, which has
The answer is (e)











Let X be a random variable with mean μx and variance σx². Let a, and b be constants.





Let W = aX + b





The mean of W, E(W) = μw is:





E(W) = E(aX + b)


= E(aX) + b


= aE(X) + b





This is true be expectation is a linear operation.





The variance of W is:





Var(W)


= Var(aX + b)


= Var(aX) {because additive constants do not affect the spread of the data}





= a²Var(X)


= (aσx)²








Here we have:





W = 0.50 * X + 3.00





where W is the cost for the car and X is the number of people in the car.





E(W) = 0.5 * 2.7 + 3.00 = 4.35





Var(W) = 0.5^2 * 1.2 = 0.30
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