Section 5.5 Summary
Subsection 5.5.1 Glossary
- Binomial Random Variable
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The number of successes among \(n\) repeats of independent trials with a probability \(p\) of success in each trial. The distribution is marked as \(\mathrm{Binomial}(n,p)\text{.}\)
- Poisson Random Variable
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An approximation to the number of occurrences of a rare event, when the expected number of events is \(\lambda\text{.}\) The distribution is marked as \(\mathrm{Poisson}(\lambda)\text{.}\)
- Density
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Histogram that describes the distribution of a continuous random variable. The area under the curve corresponds to probability.
- Uniform Random Variable
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A model for a measurement with equally likely outcomes over an interval \([a,b]\text{.}\) The distribution is marked as \(\mathrm{Uniform}(a,b)\text{.}\)
- Exponential Random Variable
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A model for times between events. The distribution is marked as \(\mathrm{Exponential}(\lambda)\text{.}\)
Subsection 5.5.2 Discuss in the Forum
This unit deals with two types of discrete random variables, the Binomial and the Poisson, and two types of continuous random variables, the Uniform and the Exponential. Depending on the context, these types of random variables may serve as theoretical models of the uncertainty associated with the outcome of a measurement.
In your opinion, is it or is it not useful to have a theoretical model for a situation that occurs in real life?
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, possibly from one of the types that are presented in this unit, can serve as a model. Discuss the importance (or lack thereof) of having a theoretical model for the situation.
For example, the Exponential distribution may serve as a model for the time until an atom of a radio active element decays by the release of subatomic particles and energy. The decay activity is measured in terms of the number of decays per second. This number is modeled as having a Poisson distribution. Its expectation is the rate of the Exponential distribution. For the radioactive element Carbon-14 (\({}^{\mbox{\tiny 14}}\mathrm{C}\)) the decay rate is \(3.8394 \times 10^{-12}\) particles per second. Computations that are based on the Exponential model may be used in order to date ancient specimens.
Subsection 5.5.3 Summary of Formulas
- Discrete Random Variable
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\begin{align*} \begin{aligned} \Expec(X) &= \sum_x \big(x \times \Prob(x)\big) \\ \Var(X) &= \sum_x\big( (x-\Expec(X))^2 \times \Prob(x)\big) \end{aligned} \end{align*}
- Continuous Random Variable
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\begin{align*} \begin{aligned} \Expec(X) &= \int \big(x \times f(x)\big)dx \\ \Var(X) &= \int\big((x-\Expec(X))^2 \times f(x) \big) dx \end{aligned} \end{align*}
- Binomial
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\(\displaystyle \Expec(X) = n p \;, \quad \Var(X) = n p(1-p)\)
- Poisson
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\(\displaystyle \Expec(X) = \lambda\;, \quad \Var(X) = \lambda\)
- Uniform
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\(\displaystyle \Expec(X) = (a+b)/2\;, \quad \Var(X)= (b-a)^2/12\)
- Exponential
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\(\displaystyle \Expec(X) = 1/\lambda\;, \quad \Var(X)= 1/\lambda^2\)
