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Analysis of Normal data 1: known population variance

Analysis of normal data

Suppose we observe one sample, which we denote \(y\) from a normal population, and we are interested in determining what that single sample can tell us about \(\mu\), the mean of the population. For this article, we will assume, implausibly, that we know the true population variance, \(\sigma^2\). We will do away with that assumption in a subsequent article, but assuming that we know the true value of \(\sigma^2\) will allow us to concentrate on learning about the population mean \(\mu\).

We begin by stating Bayes' theorem:

"The posterior is proportional to the likelihood times the prior."

or, in symbols, \[p(\mu\mid y) \propto p(y\mid\mu)p(\mu).\] The likelihood, as explained previously, is the term expressing the uncertainly inherent in the data if we knew the true value of the population mean. If our data \(y\) are normal, then we know that the likelihood is proportional to the density function of the normal distribution, which is \[p(y\mid\mu) \propto \frac{1}{\sqrt{\sigma^2}}\exp\left\{-\frac{1}{2\sigma^2}(y - \mu)^2\right\}.\] To see how the distribution of the data \(y\) is related to the population mean \(\mu\) and the variance \(\sigma^2\), you can interact with the normal distribution.

It will help us from this point onward to make a small change in the way we discuss the normal distribution. We typically describe the normal distribution in terms of its mean and variance, with the variance being a measure of the spread of the distribution. The greater the variance, the more spread out the distribution. It is also possible, however, to describe the normal distribution in terms of how precise the distribution is. A highly precise normal distribution is one in which all observations will be close to the mean; an imprecise normal is one where observations are far from the mean. We can formalize this by defining the precision of the normal distribution, which we'll call \(\tau\), as the inverse of the variance, or \[\tau=\frac{1}{\sigma^2}. To see how changing \(\tau\) affects the normal distribution, interact with a normal defined by the precision.  

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