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Ch. 4 - Discrete Probability Distributions
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Larson 8th EditionCh. 4 - Discrete Probability Distributions문제 4.1.3
4장, 문제 4.1.3

Is the expected value of the probability distribution of a random variable always one of the possible values of x? Explain.v

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Understand the concept of expected value: The expected value (E[X]) of a random variable X is a weighted average of all possible values of X, where the weights are the probabilities associated with each value. It is calculated using the formula: E[X] = Σ (xᵢ * P(xᵢ)), where xᵢ represents the possible values of X and P(xᵢ) is the probability of xᵢ.
Recognize that the expected value is not necessarily one of the possible values of X: Since the expected value is a weighted average, it can be a value that does not correspond to any specific xᵢ in the probability distribution. For example, if the probabilities and values are distributed in such a way that the average lies between two possible values, the expected value will not match any specific xᵢ.
Consider an example to illustrate: Suppose a random variable X has possible values {1, 2, 3} with probabilities {0.2, 0.5, 0.3}. The expected value is calculated as E[X] = (1 * 0.2) + (2 * 0.5) + (3 * 0.3). The result may not be exactly 1, 2, or 3, but rather a value in between.
Understand the implications: The expected value represents the 'center' or 'balance point' of the probability distribution, but it does not have to correspond to an actual observed value of the random variable. This is a key distinction between the expected value and the mode or median, which may correspond to actual observed values.
Conclude with the answer: No, the expected value of a probability distribution is not always one of the possible values of X. It depends on the specific probabilities and values in the distribution, and it is often a value that lies between the possible values of X.

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주요 개념

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Expected Value

The expected value of a random variable is a measure of the central tendency of its probability distribution. It is calculated as the weighted average of all possible values, where each value is weighted by its probability of occurrence. The expected value provides a single summary statistic that represents the long-term average outcome of a random process.
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Expected Value (Mean) of Random Variables

Probability Distribution

A probability distribution describes how the probabilities of a random variable are distributed across its possible values. It can be represented in various forms, such as a probability mass function for discrete variables or a probability density function for continuous variables. Understanding the shape and characteristics of a probability distribution is crucial for interpreting the expected value and other statistical measures.
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06:39
Calculating Probabilities in a Binomial Distribution

Random Variable

A random variable is a numerical outcome of a random phenomenon, which can take on different values based on the outcome of a random process. Random variables can be classified as discrete, taking on specific values, or continuous, taking on any value within a range. The expected value of a random variable does not have to be one of its possible values; it can be a theoretical average that may not correspond to any actual outcome.
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Intro to Random Variables & Probability Distributions
관련 실천
교과서 질문

In Exercises 1–4, find the indicated probability using the geometric distribution.

Find P(3) when p = 0.65

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교과서 질문

Finding Binomial Probabilities In Exercises 19–26, find the indicated probabilities. If convenient, use technology or Table 2 in Appendix B.


Responsible Consumption Forty-five percent of consumers say it is important that the clothing they buy is made without child labor. You randomly select 16 consumers. Find the probability that the number of of consumers who say it is important that the clothing they buy is made without child labor is (a) e

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교과서 질문

Graphical Analysis In Exercises 3–5, the histogram represents a binomial distribution with five trials. Match the histogram with the appropriate probability of success p. Explain your reasoning.

a. p = 0.25

b. p = 0.50

c. p = 0.75

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교과서 질문

Mean, Variance, and Standard Deviation In Exercises 11–14, find the mean, variance, and standard deviation of the binomial distribution with the given values of n and p.

n = 50, p = 0.4

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교과서 질문

"True or False? In Exercises 5–8, determine whether the statement is true or false. If it is false, rewrite it as a true statement.


The mean of the random variable of a probability distribution describes how the outcomes vary."

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