A business executive, transferred from Chicago to Atlanta,...

A business executive, transferred from Chicago to Atlanta, needs to sell a house in Chicago quickly. The executive's employer has offered to buy the house for $\$ 210,000$, but the offer expires at the end of the week. The executive does not currently have a better offer, but can afford to leave the house on the market for another month. From conversations with a realtor, the executive believes that leaving the house on the market for another month will bring a price that is uniformly distributed between $\$ 200,000$ and $\$ 225,000$ a. If the executive leaves the house on the market for another month, what is the mathematical expression for the probability density function of the sales price? b. If the executive leaves it on the market for another month, what is the probability the sales price will be at least $\$ 215,000 ?$ c. If the executive leaves it on the market for another month, what is the probability the sales price will be less than $\$ 210,000 ?$ d. Should the executive leave the house on the market for another month? Why or why not?

A business executive, transferred from Chicago to Atlanta, needs to sell a house in Chicago quickly. The executive's employer has offered to buy the house for $\$ 210,000$, but the offer expires at the end of the week. The executive does not currently have a better offer, but can afford to leave the house on the market for another month. From conversations with a realtor, the executive believes that leaving the house on the market for another month will bring a price that is uniformly distributed between $\$ 200,000$ and $\$ 225,000$
a. If the executive leaves the house on the market for another month, what is the mathematical expression for the probability density function of the sales price?
b. If the executive leaves it on the market for another month, what is the probability the sales price will be at least $\$ 215,000 ?$
c. If the executive leaves it on the market for another month, what is the probability the sales price will be less than $\$ 210,000 ?$
d. Should the executive leave the house on the market for another month? Why or why not?

Key Concepts

Expected Value

The expected value is the average outcome of a random variable if an experiment is repeated many times, calculated by weighting all possible outcomes by their probabilities. This concept is crucial in evaluating the attractiveness of uncertain prospects by comparing them to a sure alternative.

Integration for Continuous Probabilities

In continuous probability distributions, probabilities are computed by integrating the probability density function over the interval of interest. This integration yields the probability that the random variable falls within that particular range, making it an essential technique in the analysis of continuous probability models.

Decision Making Under Uncertainty

Decision making under uncertainty involves choosing between immediate, known options and uncertain future outcomes. It requires considering the risks, probabilities, expected values, and potential rewards associated with each decision, helping to determine whether to accept a guaranteed offer or wait for a possibly better, but uncertain, result.

Uniform Distribution

A uniform distribution is one in which every outcome in the given interval is equally likely. This means that the probability density function is constant over the interval where the distribution is defined, making it straightforward to calculate probabilities by considering the length of subintervals relative to the entire interval.

Probability Density Function (PDF)

The probability density function is a function that describes the likelihood of a continuous random variable assuming values within a specific range. For a uniform distribution, the pdf is constant over the interval, and its value is determined by ensuring that the total area under the curve over the interval equals one.

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