Section 6.5 Summary
Subsection 6.5.1 Glossary
- Normal Random Variable
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A bell-shaped distribution that is frequently used to model a measurement. The distribution is marked with \(\mathrm{Normal}(\mu,\sigma^2)\text{.}\)
- Standard Normal Distribution
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The \(\mathrm{Normal}(0,1)\text{.}\) The distribution of standardized Normal measurement.
- Percentile
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Given a percent \(p \cdot 100\%\) (or a probability \(p\)), the value \(x\) is the percentile of a random variable \(X\) if it satisfies the equation \(P(X \leq x) = p\text{.}\)
- Normal Approximation of the Binomial
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Approximate computations associated with the Binomial distribution with parallel computations that use the Normal distribution with the same expectation and standard deviation as the Binomial.
- Poisson Approximation of the Binomial
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Approximate computations associated with the Binomial distribution with parallel computations that use the Poisson distribution with the same expectation as the Binomial.
Subsection 6.5.2 Discuss in the Forum
Mathematical models are used as tools to describe reality. These models are supposed to characterize the important features of the analyzed phenomena and provide insight. Random variables are mathematical models of measurements. Some people claim that there should be a perfect match between the mathematical characteristics of a random variable and the properties of the measurement it models. Other claim that a partial match is sufficient. What is your opinion?
When forming your answer to this question you may give an example of a situation from you own field of interest for which a random variable can serve as a model. Identify discrepancies between the theoretical model and actual properties of the measurement. Discuss the appropriateness of using the model in light of these discrepancies.
Consider, for example, testing IQ. The score of many IQ tests are modeled as having a Normal distribution with an expectation of 100 and a standard deviation of 15. The sample space of the Normal distribution is the entire line of real numbers, including the negative numbers. In reality, IQ tests produce only positive values.
