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Chapter 8: Problem 59

Give an example of a real-life situation that can be modeled by a random variable with a probability distribution whose histogram is highest on the left.

Short Answer

Expert verified
An example of a real-life situation that can be modeled by a random variable with a probability distribution whose histogram is highest on the left is the income distribution in a community or country. We can define the random variable X as the income of a randomly selected individual within the community. The probability distribution of X is right-skewed, with most individuals in the lower income brackets (left side) and a long tail extending towards the higher income brackets (right side). Some properties of this distribution include a mean shifted to the right of the median and a mode representing the most common or near-lowest income bracket.

Step by step solution

01

Identify a real-life situation with a right-skewed distribution

An example of such a situation is the income distribution in a community or country. In general, a higher number of people have a lower income, while a small portion of individuals has significantly higher income. This data is typically right-skewed because most people earn in the lower income bracket, while a few outliers have extremely high income resulting in a long tail extending towards the right of the distribution.
02

Define the random variable

Let's define the random variable X as the income of a randomly selected individual within the community.
03

Describe the probability distribution

The probability distribution of X represents the likelihood of each income level for a randomly selected individual from the community. The histogram would display the frequency of each income bracket. Since it is right-skewed, most individuals would fall into the lower income brackets (left side), with fewer people in the higher income brackets (right side) and the tail extending to the right.
04

Probability distribution properties

Some properties of the probability distribution for this random variable would include: 1. The mean of the distribution would be affected by the high-income individuals, shifting it to the right of the median. 2. The median income would be lower than the mean income due to the skewness of the distribution. 3. The mode would represent the most common income bracket, which would be the lowest or close to the lowest bracket in the case of a right-skewed distribution. In conclusion, the income distribution within a community or country can be represented by a random variable with a right-skewed probability distribution, where most individuals have a lower income, and a smaller portion has extremely high income.

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