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Chapter 5: Problem 4

Statistical Literacy Consider the probability distribution of a random variable \(x .\) Is the expected value of the distribution necessarily one of the possible values of \(x\) ? Explain or give an example.

Short Answer

Expert verified
No, the expected value is not necessarily one of the possible values of \(x\), as it is a weighted average.

Step by step solution

01

Understanding the Expected Value

The expected value (also known as the mean) of a probability distribution is a weighted average of all possible values of a random variable. It is calculated by summing the product of each possible value of the random variable and its associated probability.
02

Finding the Expected Value Formula

The expected value of a discrete random variable with possible values \(x_1, x_2, \ldots, x_n\) and corresponding probabilities \(p_1, p_2, \ldots, p_n\) is computed as: \[ E(x) = x_1p_1 + x_2p_2 + \cdots + x_np_n \]
03

Considering Possibility of Non-Integral Expected Values

The expected value can be a value not present in the set of possible outcomes of the random variable. It represents an average and not necessarily an actual observed value. Therefore, it is not required to be one of the possible values of \(x\).
04

Example Illustration

Consider a random variable \(x\) that can take the values 0 and 10 with probabilities 0.5 each. The expected value is calculated as follows:\[ E(x) = 0 \times 0.5 + 10 \times 0.5 = 5 \]Notice that the expected value, 5, is not one of the possible values of \(x\).

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Key Concepts

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

Probability Distribution
A probability distribution describes how the probabilities are assigned to all possible values of a random variable. It acts like a complete guidebook for predicting the outcomes that a random variable can have. Imagine it as a detailed map that shows which routes (values) the random variable might take and how likely each one is.

Probability distributions can be discrete or continuous. Discrete distributions, like in the given exercise, deal with countable outcomes such as the roll of a dice or number of barks by a dog. Continuous distributions, on the other hand, handle outcomes that can take any value within a range, like the height of a person.
  • Possible outcomes: These are the potential values a random variable can achieve. For a discrete variable, these outcomes are finite or countably infinite.
  • Associated probabilities: Each outcome has a probability which explains how likely it is to occur. These probabilities must add up to 1.
Random Variable
A random variable is a fundamental concept in probability and statistics. It is essentially a function that assigns a numerical value to each outcome in a sample space. Random variables help us model uncertainties and assign numbers to possible outcomes.

There are two types of random variables:
  • Discrete Random Variable: As in our exercise, it takes specific, isolated values. For instance, the example random variable can take values of 0 or 10.
  • Continuous Random Variable: It can take any value within a continuous range. For example, temperatures or distances often represent continuous random variables.
Random variables are usually denoted by letters like x, y, or z, and they provide the framework for developing probability distributions.
Statistical Literacy
Statistical literacy refers to the ability to interpret, understand and critically evaluate statistical information. Being statistically literate means that you can make sense of numbers and data presented in studies, reports, and real-life situations.

In our exercise, statistical literacy involves understanding how expected value works and recognizing that it is a weighted average that does not need to be a possible value of the random variable itself. This helps us see beyond raw numbers to the underlying patterns and insights.
  • Expected Value: A key concept, it represents the long-term average of repetitions of an experiment it would yield.
  • Interpreting results: Being able to distinguish between actual possible outcomes and theoretical averages permits better decision-making.
Statistical literacy empowers individuals to make informed decisions based on reliable data analysis, thus playing a crucial role in numerous fields like business, science, and policy-making.

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Most popular questions from this chapter

Conditional Probability: Hail Damage In western Kansas, the summer density of hailstorms is estimated at about \(2.1\) storms per 5 square miles. In most cases, a hailstorm damages only a relatively small area in a square mile (Reference: Agricultural Statistics, U.S. Department of Agriculture). A crop insurance company has insured a tract of 8 square miles of Kansas wheat land against hail damage. Let \(r\) be a random variable that represents the number of hailstorms this summer in the 8 -square-mile tract. (a) Explain why a Poisson probability distribution is appropriate for \(r\). What is \(\lambda\) for the 8 -square-mile tract of land? Round \(\lambda\) to the nearest tenth so that you can use Table 4 of Appendix II for Poisson probabilities. (b) If there already have been two hailstorms this summer, what is the probability that there will be a total of four or more hailstorms in this tract of land? Compute \(P(r \geq 4 \mid r \geq 2)\). (c) If there already have been three hailstorms this summer, what is the probability that there will be a total of fewer than six hailstorms? Compute \(P(r<6 \mid r \geq 3)\).

Critical Thinking Let \(r\) be a binomial random variable representing the number of successes out of \(n\) trials. (a) Explain why the sample space for \(r\) consists of the set \(\\{0,1,2, \ldots, n\\}\) and why the sum of the probabilities of all the entries in the entire sample space must be 1 . (b) Explain why \(P(r \geq 1)=1-P(0)\). (c) Explain why \(P(r \geq 2)=1-P(0)-P(1)\). (d) Explain why \(P(r \geq m)=1-P() 0-P(1)-\cdots-P(m-1)\) for \(1 \leq m \leq n\).

Sociology: Dress Habits A research team at Cornell University conducted a study showing that approximately \(10 \%\) of all businessmen who wear ties wear them so tightly that they actually reduce blood flow to the brain, diminishing cerebral functions (Source: Chances: Risk and Odds in Everyday Life, by James Burke). At a board meeting of 20 businessmen, all of whom wear ties, what is the probability that (a) at least one tie is too tight? (b) more than two ties are too tight? (c) no tie is too tight? (d) at least 18 ties are not too tight?

Law Enforcement: Property Crime Does crime pay? The FBI Standard Survey of Crimes shows that for about \(80 \%\) of all property crimes (burglary, larceny, car theft, etc.), the criminals are never found and the case is never solved (Source: True Odds, by James Walsh, Merrit Publishing). Suppose a neighborhood district in a large city suffers repeated property crimes, not always perpetuated by the same criminals. The police are investigating six property crime cases in this district. (a) What is the probability that none of the crimes will ever be solved? (b) What is the probability that at least one crime will be solved? (c) What is the expected number of crimes that will be solved? What is the standard deviation? (d) Quota Problem How many property crimes \(n\) must the police investigate before they can be at least \(90 \%\) sure of solving one or more cases?

Sociology: Hawaiians On the leeward side of the island of Oahu, in the small village of Nanakuli, about \(80 \%\) of the residents are of Hawaiian ancestry (Source: The Honolulu Advertiser). Let \(n=1,2,3, \ldots\) represent the number of people you must meet until you encounter the first person of Hawaiian ancestry in the village of Nanakuli. (a) Write out a formula for the probability distribution of the random variable \(n\). (b) Compute the probabilities that \(n=1, n=2\), and \(n=3\). (c) Compute the probability that \(n \geq 4\). (d) In Waikiki, it is estimated that about \(4 \%\) of the residents are of Hawaiian ancestry. Repeat parts (a), (b), and (c) for Waikiki.
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