Section 4.7 Summary
Subsection 4.7.1 Glossary
- Random Variable
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The probabilistic model for the value of a measurement, before the measurement is taken.
- Sample Space
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The set of all values a random variable may obtain.
- Probability
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A number between 0 and 1 which is assigned to a subset of the sample space. This number indicates the likelihood of the random variable obtaining a value in that subset.
- Expectation
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The central value for a random variable. The expectation of the random variable \(X\) is marked by \(\Expec(X)\text{.}\)
- Variance
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The (squared) spread of a random variable. The variance of the random variable \(X\) is marked by \(\Var(X)\text{.}\) The standard deviation is the square root of the variance.
Subsection 4.7.2 Discussion in the Forum
Random variables are used to model situations in which the outcome, before the fact, is uncertain. One component in the model is the sample space. The sample space is the list of all possible outcomes. It includes the outcome that took place, but also all other outcomes that could have taken place but never did materialize. The rationale behind the consideration of the sample space is the intention to put the outcome that took place in context. What do you think of this rationale?
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 the sample space for that random variable and discuss the importance (or lack thereof) of the correct identification of the sample space.
For example, consider a factory that produces car parts that are sold to car makers. The role of the QA personnel in the factory is to validate the quality of each batch of parts before the shipment to the client.
To achieve that, a sample of parts may be subject to a battery of quality test. Say that 20 parts are selected to the sample. The number of those among them that will not pass the quality testing may be modeled as a random variable. The sample space for this random variable may be any of the numbers 0, 1, 2, …, 20.
The number 0 corresponds to the situation where all parts in the sample passed the quality testing. The number 1 corresponds to the case where 1 part did not pass and the other 19 did. The number 2 describes the case where 2 of the 20 did not pass and 18 did pass, etc.
Subsection 4.7.3 Summary of Formulas
- Population Size
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\(N\) = the number of people, things, etc. in the population.
- Population Average
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\(\displaystyle \mu = (1/N)\sum_{i=1}^N x_i\)
- Expectation of a Random Variable
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\(\displaystyle \Expec(X) = \sum_x \big(x \times \Prob(x)\big)\)
- Population Variance
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\(\displaystyle \sigma^2 = (1/N)\sum_{i=1}^N (x_i-\mu)^2\)
- Variance of a Random Variable
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\(\displaystyle \Var(X) = \sum_x\big( (x-\Expec(X))^2 \times \Prob(x)\big)\)
