Question

Consider a large ferry that can accommodate cars and buses. The toll for cars is $$\$ 3$$, and the toll for buses is $$\$ 10 .$$ Let \(x\) and \(y\) denote the number of cars and buses, respectively, carried on a single trip. Cars and buses are accommodated on different levels of the ferry, so the number of buses accommodated on any trip is independent of the number of cars on the trip. Suppose that \(x\) and \(y\) have the following probability distributions: $$ \begin{array}{lrrrrrr} x & 0 & 1 & 2 & 3 & 4 & 5 \\ p(x) & .05 & .10 & .25 & .30 & .20 & .10 \\ y & 0 & 1 & 2 & & & \\ p(y) & .50 & .30 & .20 & & & \end{array} $$ a. Compute the mean and standard deviation of \(x\). b. Compute the mean and standard deviation of \(y\). c. Compute the mean and variance of the total amount of money collected in tolls from cars. d. Compute the mean and variance of the total amount of money collected in tolls from buses. e. Compute the mean and variance of \(z=\) total number of vehicles (cars and buses) on the ferry. f. Compute the mean and variance of \(w=\) total amount of money collected in tolls.

Short answer

Detailed calculations are required to compute statistical measures. However, the steps described above give a clear path for doing so. Once you follow these steps and compute the required values, both mean and standard deviation for each case could be obtained. And mean will give you an idea of expected value while standard deviation will provide a measure of the dispersion.

Step-by-step solution

Compute the Mean and Standard Deviation of x and y

Mean (\(E[x]\) or \(E[y]\)) is computed by multiplying each possible outcome by its probability and then summing these products. Standard deviation (\(\sigma_{x}\) or \(\sigma_{y}\)) is the square root of the variance, where variance (\(\sigma_{x}^{2}\) or \(\sigma_{y}^{2}\)) is calculated as the expectation of the squared deviation of a random variable from its mean.

Compute the Mean and Variance of Revenue from Cars and Buses

The mean revenue (\(E[R_x]\) or \(E[R_y]\)) is computed by multiplying the mean number of cars or buses with their respective tolls. Variance (\(\sigma_{R_x}^{2}\) or \(\sigma_{R_y}^{2}\)) is calculated as \(Toll^{2}\) times the variance of the number of cars or buses.

Compute the Mean and Variance of Total Number of Vehicles on the Ferry

The total number of vehicles 'z' is the sum of the number of cars 'x' and buses 'y'. The mean of 'z' (\(E[z]\)) is the sum of the means of 'x' and 'y'. Because 'x' and 'y' are independent, the variance of 'z' (\(\sigma_{z}^{2}\)) is the sum of variances of 'x' and 'y'.

Compute the Mean and Variance of Total Revenue Collected

The total revenue 'w' is the sum of the revenues from cars and buses. The mean of 'w' (\(E[w]\)) is the sum of the means of 'R_x' and 'R_y'. Because the revenues from cars and buses are independent, the variance of 'w' (\(\sigma_{w}^{2}\)) is the sum of variances of 'R_x' and 'R_y'.