91Ó°ÊÓ

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.

Step by step solution

01

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.

02

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.

03

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'.

04

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'.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean and Standard Deviation

When dealing with probability distributions, the mean and standard deviation provide insight into the distribution's central tendency and spread, respectively.

The mean, also known as the expected value, is a measure of the average outcome you would expect if you were to repeat the random process numerous times. To compute the mean (\(E[x]\) or \(E[y]\)), you multiply each possible outcome by its probability and sum these products:

• This provides a weighted average, taking into account all possible outcomes and their likelihoods.

Once you have the mean, you can also calculate the variance. The variance measures how much the values of the random variable deviate from the mean on average.

However, since variance is in squared units, it might not be immediately interpretable in the context of your original data. Thus, we often use the standard deviation, which is the square root of the variance, to describe the spread of a distribution in the same units as the original data. This measure gives a more intuitive sense of variability by quantifying how much the values typically differ from the mean.

Variance

Variance (\(\sigma_x^2 \) or \(\sigma_y^2 \)) quantifies the dispersion of a set of values. It describes how far each number in the set is from the mean and thus from each other.

To calculate the variance of a discrete random variable such as the number of cars (\(x\)) or buses (\(y\)), follow these steps:

• First, find the mean of the data (expected value).
• Then, subtract the mean from each data point to find the deviation of each point from the mean.
• Square each of these deviations to ensure they are positive and to give more weight to larger deviations.
• Finally, multiply each squared deviation by its respective probability, and sum these values to get the variance.

While variance helps quantify variability, comparing variances directly may not be useful if the units are squared. Thus, using standard deviation becomes a common alternative, as it is expressed in the same units as the data, making it more interpretable in the context of the problem.

Random Variables

Random variables are a fundamental concept in probability theory used to model random processes or experiments. A random variable \(x\) or \(y\) assigns a number to each outcome in a sample space.

Consider the ferry problem where \(x\) represents the number of cars and \(y\) the number of buses:

• Here, both are discrete random variables, meaning they can only take on certain specified values, unlike continuous random variables, which can take any value within a range.
• The probability distribution of each random variable specifies the probability of each possible value.

Additionally, random variables can be independent. In the ferry problem, the numbers of cars and buses are independent.

This means that the outcome of one variable doesn't affect the other, simplifying the calculation of combined measures such as total vehicles or total revenue.