
Contents
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What is a function?
What is a function?
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Can you give me an example of a continuous random variable?
Can you give me an example of a continuous random variable?
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What is the probability that a bacterium lives exactly 5 hours?
What is the probability that a bacterium lives exactly 5 hours?
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So, what can we do?
So, what can we do?
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Can we see an example of a probability density function?
Can we see an example of a probability density function?
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I see…and what distribution would result from this pdf?
I see…and what distribution would result from this pdf?
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Why is the density 0.5 in this example?
Why is the density 0.5 in this example?
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Can we formally define a uniform pdf?
Can we formally define a uniform pdf?
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What is the probability that x is between 4.5 and 5.5 hours for our uniform distribution?
What is the probability that x is between 4.5 and 5.5 hours for our uniform distribution?
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What is the probability that x is exactly 5 hours?
What is the probability that x is exactly 5 hours?
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Are there other examples of continuous probability density functions?
Are there other examples of continuous probability density functions?
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What exactly is the normal distribution?
What exactly is the normal distribution?
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And what does “Gaussian” refer to?
And what does “Gaussian” refer to?
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What does a normal (Gaussian) distribution look like?
What does a normal (Gaussian) distribution look like?
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So, the normal distribution has two parameters?
So, the normal distribution has two parameters?
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How were these distributions generated?
How were these distributions generated?
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What exactly is the normal (Gaussian) pdf?
What exactly is the normal (Gaussian) pdf?
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Can you give me an example of how to use the normal pdf?
Can you give me an example of how to use the normal pdf?
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So, we plug in multiple values for x and generate the distribution?
So, we plug in multiple values for x and generate the distribution?
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How do we go from probability density to probability with a normal pdf?
How do we go from probability density to probability with a normal pdf?
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What is the probability that a bacterium has a lifespan between 4.5 and 5.5 hours?
What is the probability that a bacterium has a lifespan between 4.5 and 5.5 hours?
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Is the total area equal to 1.0?
Is the total area equal to 1.0?
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How would one express this mathematically?
How would one express this mathematically?
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With 41 slivers, what is the probability that a bacterium has a lifespan between 4.5 and 5.5 hours?
With 41 slivers, what is the probability that a bacterium has a lifespan between 4.5 and 5.5 hours?
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What is the total area if our rectangles became really, really skinny?
What is the total area if our rectangles became really, really skinny?
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Do all probability density functions have an area under the curve = 1.0?
Do all probability density functions have an area under the curve = 1.0?
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What would the integral look like for the normal pdf?
What would the integral look like for the normal pdf?
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OK, one more time! What is the probability that a bacterium has a lifespan between 4.5 and 5.5 hours?
OK, one more time! What is the probability that a bacterium has a lifespan between 4.5 and 5.5 hours?
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How does one go about integrating?
How does one go about integrating?
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Are there refresher courses that review this material?
Are there refresher courses that review this material?
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What other probability density functions are there?
What other probability density functions are there?
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Likelihood Likelihood
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OK, what exactly is likelihood?
OK, what exactly is likelihood?
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What if you don’t know that σ is 0.5?
What if you don’t know that σ is 0.5?
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How can this be used in a Bayesian inference problem?
How can this be used in a Bayesian inference problem?
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Can you estimate the probability of a specific hypothesis for theta?
Can you estimate the probability of a specific hypothesis for theta?
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So, there are no specific hypotheses?
So, there are no specific hypotheses?
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Can we see a Bayesian inference problem with infinite hypotheses?
Can we see a Bayesian inference problem with infinite hypotheses?
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Can we depict this problem graphically?
Can we depict this problem graphically?
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If we couldn’t integrate the normal distribution, how on earth are we going to integrate the denominator of Bayes’ Theorem?
If we couldn’t integrate the normal distribution, how on earth are we going to integrate the denominator of Bayes’ Theorem?
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Cite
Abstract
This chapter builds on probability distributions. Its focus is on general concepts associated with probability density functions (pdf’s), which are distributions associated with continuous random variables. The continuous uniform and normal distributions are highlighted as examples of pdf’s. These and other pdf’s can be used to specify prior distributions, likelihoods, and/or posterior distributions in Bayesian inference. Although this chapter specifically focuses on the continuous uniform and normal distributions, the concepts discussed in this chapter will apply to other continuous probability distributions. By the end of the chapter, the reader should be able to define and use the following terms for a continuous random variable: random variable, probability distribution, parameter, probability density, likelihood, and likelihood profile.
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