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A contractor has submitted bids on three state jobs: an office building, a theater, and a parking garage. State rules do not allow a contractor to be offered more than one of these jobs. If this contractor is awarded any of these jobs, the profits earned from these contracts are \(\$ 10\) million from the office building, \(\$ 5\) million from the theater, and \(\$ 2\) million from the parking garage. His profit is zero if he gets no contract. The contractor estimates that the probabilities of getting the office building contract, the theater contract, the parking garage contract, or nothing are \(.15, .30, .45\), and \(.10\), respectively. Let \(x\) be the random variable that represents the contractor's profits in millions of dollars. Write the probability distribution of \(x .\) Find the mean and standard deviation of \(x\). Give a brief interpretation of the values of the mean and standard deviation.

Step by step solution

01

Identify the values of the random variable and their respective probabilities

The random variable \(x\) can have the values 0, 2, 5, and 10, corresponding to the possible profits in millions of dollars from no contract, the parking garage contract, the theater contract, and the office building contract, respectively. The respective probabilities of these events occur are 0.10, 0.45, 0.30 and 0.15.

02

Calculate the mean of the random variable

The mean or expected value of a random variable is found by multiplying each possible value of the variable by its respective probability and then summing up these products. Therefore, the mean \( \mu \) is calculated as: \( \mu = (0*0.10) + (2*0.45) + (5*0.30) + (10*0.15) \).

03

Calculate the standard deviation of the random variable

The standard deviation of a random variable is the square root of its variance. The variance is calculated as the sum of the squared difference between each value and the mean, multiplied by the respective probabilities. Therefore, variance \( \sigma^2 \) is calculated as: \( \sigma^2 = (0- \mu)^2*0.10 + (2- \mu)^2*0.45 + (5- \mu)^2*0.30 + (10- \mu)^2*0.15 \), where \( \mu \) is the mean obtained from Step 2, and the standard deviation \( \sigma \) is \(\sqrt{\sigma^2} \).

04

Interpret the mean and standard deviation

The mean value represents the average income that the contractor can expect if he repeats the same bidding process many times. The standard deviation indicates how spread out the profits can be around this expected value. A large standard deviation means that the outcome has higher variability.

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Key Concepts

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

Mean of a Random Variable

The mean of a random variable is essentially the average of all possible outcomes, but it's not just any ordinary average. When dealing with random variables like the contractor's potential profits, the mean is calculated using a probability-weighted method.
This means each possible value (\(x\)) that the contractor might achieve is multiplied by its probability of occurrence, and all these products are then summed up to give us the mean.

• The mean is often denoted by the symbol \( \mu \).
• In our example: \( \mu = (0*0.10) + (2*0.45) + (5*0.30) + (10*0.15) \).

This calculation results in the expected average outcome in the long run if the contractor were to bid over and over again under similar circumstances. Understanding this helps in predicting outcomes and making informed decisions.

Standard Deviation

Standard deviation is a statistical tool used to measure the amount of variation or dispersion in a set of values. In the case of the contractor's profits, it helps us understand how much the potential earnings can deviate from the mean.
First, we have to calculate variance, which involves the sum of the squared differences between each possible profit value and the mean, weighted by their probabilities.

• The variance \( \sigma^2 \) is given by: \( \sigma^2 = (0- \mu)^2*0.10 + (2- \mu)^2*0.45 + (5- \mu)^2*0.30 + (10- \mu)^2*0.15 \).

Standard deviation \( \sigma \) is simply the square root of this variance.

• Mathematically, \( \sigma = \sqrt{\sigma^2} \).

A smaller standard deviation indicates that the results are more clustered around the mean, whereas a larger one suggests more variation. This can be crucial for assessing risk and variability in potential profits.

Expected Value

The expected value of a random variable is a central concept in probability that gives us the long-term average or expected outcome of a random event. It is similar to the mean but specifically in the context of random variables.
In essence, it is the sum of all possible values each multiplied by their respective probability.

• For the contractor, the expected value reflects the average profit expected if similar bidding situations were repeated multiple times.
• The formula: \( E(x) = \sum x_i p_i \) aligns with the mean calculation in our specific problem.

This expected value helps in making forecasts and decisions, guiding the contractor to understand what average profits might be expected in the long term, even though actual results can vary each time.

Variance

Variance is a statistical measure that shows how much a set of values diverges from the average of the values.
It provides a numerical value to express how much individual numbers in a set differ from the mean of the set, and thus helps in understanding the spread of a distribution.

• The variance is calculated by taking the average of the squared differences from the Mean.
• In our example, variance gives insight into the variability of the contractor's potential profits.

Mathematically, the variance \( \sigma^2 \) is calculated as \( \sigma^2 = \sum (x_i - \mu)^2 p_i \), where \( x_i \) are possible values, \( \mu \) is the mean, and \( p_i \) are the probabilities.
A higher variance indicates a larger dispersion of possible values from the mean, which can imply more risk but also more potential for diverse outcomes.