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Chapter 8: Problem 58

What do you expect to happen to the probabilities in a probability distribution as you make the measurement classes smaller?

Short Answer

Expert verified
As we make the measurement classes smaller in a probability distribution, the probabilities associated with each class will decrease since the total probability of 1 is now divided among a larger number of classes. This results in a more refined distribution with narrower intervals, allowing for more precision in understanding the distribution of the random variable. However, the overall shape of the probability distribution curve remains roughly the same, with the total probability distribution equal to 1.

Step by step solution

01

Understanding Probability Distributions

A probability distribution is a function that describes the likelihood of different possible outcomes for a random variable or event. It assigns probabilities to each possible value or range of values that the random variable can take. The sum of all probabilities in a probability distribution equals 1. There are two types of probability distributions: discrete and continuous. Discrete probability distributions represent events or outcomes that take a finite number of values; continuous probability distributions represent events or outcomes that can take on an infinite number of values within a certain range.
02

Understanding Measurement Classes

Measurement classes are the intervals or categories within which data is collected and analyzed. In a probability distribution, these intervals represent the possible outcomes or ranges of values for the random variable. For example, in a dice roll experiment, measurement classes could be the possible outcomes (1, 2, 3, 4, 5, and 6), or, within a continuous probability distribution, they could be specified intervals such as measuring the height of people in intervals of 3 inches (e.g., 5'0"-5'3", 5'3"-5'6", etc.).
03

Making Measurement Classes Smaller

As we make the measurement classes smaller, we are essentially increasing the number of distinct intervals or categories within the probability distribution. For example, if the height intervals of people were decreased to 1-inch intervals, we would have more specific information about how people's heights are distributed. In a continuous probability distribution, this would involve reducing the size of the intervals on the x-axis of the distribution, making them narrower. In a discrete probability distribution, this would involve increasing the number of individual events or outcomes considered.
04

Analyzing the Effects on Probabilities

When the measurement classes become smaller, the probabilities associated with each class will also become smaller as we divide the total probability of 1 by a larger number of classes. This is intuitively logical because when there's more precision in defining intervals, it's less likely that a single interval contains a large share of the possible outcomes. However, the overall shape of the probability distribution curve will remain roughly the same 鈥� as the size of the classes decreases, the number of classes increases, keeping the total probability distribution equal to 1.
05

Conclusion

As the measurement classes in a probability distribution become smaller, the probabilities associated with each class decrease since the total probability of 1 is now divided among a larger number of classes. The overall shape of the probability distribution will remain roughly the same, although intervals are narrower and the distribution is more refined. This allows for a more precise understanding of the distribution of the random variable, but each smaller interval carries a lower individual probability.

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Most popular questions from this chapter

The Acme Insurance Company is launching a drive to generate greater profits, and it decides to insure racetrack drivers against wrecking their cars. The company's research shows that, on average, a racetrack driver races four times a year and has a 1 in 10 chance of wrecking a vehicle, worth an average of \(\$ 100,000\), in every race. The annual premium is \(\$ 5,000\), and Acme automatically drops any driver who is involved in an accident (after paying for a new car), but does not refund the premium. How much profit (or loss) can the company expect to earn from a typical driver in a year? HINT [Use a tree diagram to compute the probabilities of the various outcomes.]

Calculate the expected value of the given random variable \(X .\) [Exercises \(23,24,27\), and 28 assume familiarity with counting arguments and probability (Section 7.4).] Select five cards without replacement from a standard deck of 52, and let \(X\) be the number of red cards you draw.

Calculate the standard deviation of \(X\) for each probability distribution. (You calculated the expected values in the last exercise set. Round all answers to two decimal places.) $$ \begin{array}{|c|c|c|c|c|} \hline \boldsymbol{x} & 10 & 20 & 30 & 40 \\ \hline \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) & \frac{3}{10} & \frac{2}{5} & \frac{1}{5} & \frac{1}{10} \\ \hline \end{array} $$

The following figures show the price of silver per ounce, in dollars, for the 10 -business day period Feb. 2 -Feb. 13 , $$ \begin{aligned} &2009: 22 \\ &12.4,12.4,12.4,12.8,12.9,13.0,13.0,13.4,13.3,13.4 \end{aligned} $$ Find the sample mean, median, and mode(s). What do your answers tell you about the price of silver?

Calculate the expected value, the variance, and the standard deviation of the given random variable \(X .\) You calculated the expected values in the last exercise set. Round all answers to two decimal places.) \(X\) is the number of tails that come up when a coin is tossed three times.
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