# A review of sampling distributions and the univariate Gaussian

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## Outline

• The following topics will be covered in this lecture:
• Sample statistics
• Sample random variables
• Sampling distributions
• The univariate Gaussian distribution
• Properties of the univariate Gaussian
• The central limit theorem

## Sample statistics

• The goal statistics is to use a numerical summary of data from a small, representative sample to say something general about the larger, unobservable population or phenomena.

• The measures of the population are referred to as parameters.

• Parameters are generally unknown and unknowable.

• For example, we cannot exactly compute the mean sea-surface temperature globally, as it is impossible to take all such measurements.
• However, if we have a representative sample, we can compute the sample mean.

• Numerical values like the sample mean computed from data are referred to as statistics.
• The sample mean will almost surely not equal population mean, due to the natural variation (sampling error) that occurs in any given sample.

• However, if we have a good probabilistic model for the population, we can use the sample statistic to estimate the general, unknown population parameter.
• RVs and probability distributions give us the model for estimating population parameters.

• Note: we can only “find” the parameters exactly in very simple examples like games of chance.

• Generally, we will have to be satisfied with estimates of the parameters that are uncertain, but also include measures of “how uncertain”.

### Sample mean

• Suppose we have a sample of $$n$$ total measurements of some RV $$X$$.

• We will denote these measurements $$x_1, x_2, \cdots, x_n \in \mathbb{R}$$, where these refer to fixed numerical values.
• These may correspond to the value that $$X$$ attains upon $$n$$ independently replicated trials.
The (arithmetic sample) mean
Given measurements $$x_1,\cdots,x_n$$ of the RV $$X$$, we say that the sample mean is defined $\text{Sample mean} = \hat{x} = \frac{x_1 +x_2 +\cdots + x_n}{n}= \frac{\sum_{i=1}^n x_i}{n}$
• We remark that $$\hat{x}$$ is a fixed numerical value depending on the particular sequence of outcomes $$x_1,\cdots, x_n$$ observed.

• Due to this fact, with respect to a new sample of size $$n$$, we may attain a new value for the sample mean.

### Sample variance and standard deviation

• We can similarly define the sample variance and standard deviation as follows
Sample standard deviation
Given measurements $$x_1,\cdots,x_n$$ of the RV $$X$$, we say that the sample standard deviation $\hat{\sigma} = \sqrt{\frac{\sum_{i=1}^n\left(x_i - \hat{x}\right)^2}{n-1}}$
• Note that the numerator in the above accounts for the fact that one degree of freedom has been utilized in the computation of $$\hat{x}$$.
Sample variance
Given measurements $$x_1,\cdots,x_n$$ of the RV $$X$$, we say that the sample variance $\hat{\sigma}^2 = \frac{\sum_{i=1}^n\left(x_i - \hat{x}\right)^2}{n-1}$
• For the same reasons discussed for the sample mean, the sample standard deviation and variance will tend to differ depending on the particular sequence of outcomes $$x_1,\cdots, x_n$$ measured.

• This discrepancy is what we call sampling error, in which the random variation in a sample of a fixed size $$n$$ upon replication produces differences in the computation of a statistic.

• For this reason, we may also consider a probabilistic model for the sample statistic, depending on the replication of measurements.

## Sample random variables

• Specifically, suppose that we want to obtain an estimate of a population parameter, where the population is modeled with a RV $$X$$.

• We know that before the data are collected, the observations are considered to be RVs,

• i.e., we treat an independent sequence of measurements of $$X$$,

$X_1, X_2, \cdots , X_n$

• as RVs all drawn from a parent distribution $$X \sim P$$ (where the CDF will define the distribution).
Random sample
The RVs $$X_1 , X_2, \cdots , X_n$$ are a random sample of size $$n$$ if the $$X_i$$’s are independent RVs and every $$X_i$$ has the same probability distribution.
• We then say that the measurements we obtain are possible outcomes of the sample variables $$\{X_i\}_{i=1}^n$$;

• particularly, if we make a computation of the sample mean, $\hat{X} = \frac{1}{n} \sum_{i=1}^n X_i$

the above is treated as a RV (a linear combination of RVs) which has a random outcome, dependent on the realizations of the $$X_i$$.

## Sampling distributions

• More generally, any function of the observations, i.e., any statistic, is also modeled as a RV.
Point estimators
Let $$\{X_j\}_{j=1}^n$$ be a random sample. Let $$\theta$$ be a parameter of the parent population, defined by the CDF $$P$$. If $$h$$ is a general function used to compute some statistic estimating $$\theta$$, we thus define the RV $\hat{\Theta} = h(X_1, \cdots, X_n)$ to be a point estimator for $$\theta$$.
• We call the probability distribution of a statistic or estimator as above a sampling distribution.

Sampling Distribution
The probability distribution of a statistic is called a sampling distribution.
• In this framework, we will distinguish then between the estimator (a random variable) and the numerical value it might attain on a sample of measurements.

Point estimate
A point estimate of some population parameter $$\theta$$ is a single numerical value
$\hat{\theta} = h(x_1, \cdots,x_n)$ attained as a particular realization of the RV $$\hat{\Theta}$$.

### Sampling distributions

• The notion of the “center” of the sampling distribution can be useful as a general criteria for estimators.

• Formally, we say that $$\hat{\Theta}$$ is an unbiased estimator of $$\theta$$ if the expected value of $$\hat{\Theta}$$ is equal to $$\theta$$.

• This is equivalent to saying that the mean of the sampling distribution of $$\hat{\Theta}$$ is equal to $$\theta$$.

Bias of an Estimator
The point estimator $$\hat{\Theta}$$ is an unbiased estimator for the parameter $$\theta$$ if $\mathbb{E}\left[\hat{\Theta}\right] = \theta$ If the estimator is not unbiased, then the difference $\mathbb{E}\left[\hat{\Theta}\right] - \theta$ is called the bias of the estimator $$\hat{\Theta}$$. When an estimator is unbiased, the bias is zero; that is, \begin{align} \mathbb{E}\left[\hat{\Theta}\right] - \theta &= \theta - \theta \\ &=0 \end{align}
• If we consider the expected value to represent the average value over infinite replications;

• the above says that “over infinite replications of a random sample of size $$n$$, the average value of the point estimator $$\hat{\Theta}$$ will equal the true population parameter $$\theta$$”.

### Sampling distributions

• Both of the
1. sample mean $\hat{X}= \frac{1}{n}\sum_{i=1}^n X_i;$ and
2. sample variance $\hat{\sigma}^2 = \frac{\sum_{i=1}^n \left(X_i - \hat{X}\right)^2}{n-1}$
• are unbiased estimators, i.e., \begin{align} \mathbb{E}\left[\hat{X}\right] = \overline{x}, & & \mathbb{E}\left[\hat{\sigma}^2\right] = \sigma^2. \end{align}

• However, there are theoretical reasons that we can use to show that the sample standard deviation is a biased estimator of the population standard deviation, i.e.,

$\mathbb{E}\left[ \hat{\sigma}\right] \leq \sigma$

and it consistently underestimates the true standard deviation.

• The bias tends to be small, however, and it is still the most practical estimate most of the time for the population standard deviation.

### Sampling distributions

• Recalling that the expected value gives the center of mass of the probability distribution, we should also be interested in the spread of the sampling distribution.

• As noted before, the variance is a “natural” measure of spread mathematically for theoretical reasons, but it is in the units squared of the original units.

• For this reason, when we talk about the spread of an estimator's sampling distribution, we typically discuss the standard error.

The standard error
Let $$\hat{\Theta}$$ be a point estimator of $$\theta$$. The standard error error of $$\hat{\Theta}$$ is its standard deviation given by $\sigma_\hat{\Theta} = \sqrt{\mathrm{var}\left(\hat{\Theta}\right)}.$ If the standard error involves unknown parameters that can be estimated, substitution of those values into the equation above produces an estimated standard error denoted $$\hat{\sigma}_\hat{\Theta}$$. It is also common to write the standard error as $$\mathrm{SE}\left(\hat{\Theta}\right)$$.
• With these constructions in mind, we will now introduce one of the most fundamental results of classical statistics.

• This result establishes the normal or Gaussian distribution in its central importance among distributions.

## The univariate Gaussian distribution

• The Gaussian distribution is considered the most prominent distribution in statistics.
• It is a continuous probability distribution that has a bell-shaped probability density function.
• The Gaussian distribution arises from the central limit theorem (CLT),
• under weak conditions, the sum of a large number of RVs drawn from the same distribution is distributed approximately normally irrespective of the form of the original distribution.
• This gives mathematical justification to why we see normally distributed data quite often in practice; as was noted by Henri Poincare
• “Everybody believes in the exponential law of errors [i.e., the normal / Gaussian distribution]: the experimenters, because they think it can be proved by mathematics; and the mathematicians, because they believe it has been established by observation.” — Poincare, Henri “Calcul Des Probabilités.”
• In addition to the ubiquity of the normal distribution, it can be easily manipulated analytically in equations,
• this enables one to derive a large number of results in explicit form.
• Due to these two aspects, the normal distribution is used extensively in theory and practice.

### The univariate Gaussian distribution continued

• Unlike how we defined the density function $$p$$ and used this to compute $$\overline{x}$$ and $$\sigma$$ formerly, we will reverse this for the normal.
• That is, we will use $$\overline{x}$$ and $$\sigma$$ to define the density of the normal and parametrize the distribution.
• Let us use the following notation for compactness where $\exp(x) = e^{x}.$
• The univariate Gaussian distribution
Let the Gaussian RV $$X$$ have mean $$\overline{x}$$ and standard deviation $$\sigma$$. The probability density function is given as \begin{align} p(x) = \frac{1}{\sqrt{2\pi}\sigma}\exp\left(-\frac{\left(x - \overline{x}\right)^2}{2\sigma^2}\right). \end{align} We will write $$X \sim N\left(\overline{x}, \sigma^2\right)$$ to denote that $$X$$ has the density described above.
• Recall how we considered $$\overline{x}$$ to be a measure of center and $$\sigma$$ a measure of spread.
• If we vary these two values, we can change the center of mass and the spread of the normal distribution: • In the case that $$\overline{x}=0$$ and $$\sigma=1$$, we denote $$N(0, 1)$$ to be the standard normal distribution.

### The univariate Gaussian distribution continued

• Another useful property of the family of Gaussian distributions is that it is closed under linear transformations.
Closure of the Gaussian under linear transformations
Let $$X_1$$ and $$X_2$$ be independent, Gaussian RVs defined \begin{align} X_1\sim N\left(\overline{x}_1 , \sigma_1^2 \right) & & X_2 \sim N\left(\overline{x}_2, \sigma_2^2 \right). \end{align} Then for $$a,b,c \in \mathbb{R}$$, the linear combination satisfies $aX_1 + bX_2 + c \sim N\left(a \overline{x}_1 + b\overline{x}_2 + c, a^2 \sigma_1^2 + b^2 \sigma_2^2\right)$
• This is actually a general property of the family of stable distributions.

• The closure property above implies that a Gaussian variable can always be “standardized” as,

\begin{align} X \sim N(\overline{x}, \sigma^2) && \Rightarrow && \frac{X - \overline{x}}{\sigma} \sim N(0, 1). \end{align}

• The closure of the Gaussian under linear transformations has extremely important implications, when we introduce a mechanistic model later.

• This is at the basis of results for estimators defined in a class of models known as Gauss-Markov models.

## Central limit theorem

• Suppose that a random sample of size $$n$$ is taken from a normal population with mean $$\overline{x}$$ and variance $$\sigma^2$$.

• By definition of a random sample each observation in this sample, say, $$X_1, X_2, \cdots, X_n$$, is a normally and independently distributed RV with mean $$\overline{x}$$ and variance $$\sigma^2$$.

• We conclude that, due to closure of the Gaussian, the sample mean

$\hat{X}= \frac{X_1 + X_2 + \cdots + X_n}{n}$

has a normal distribution with mean

\begin{align} \mathbb{E}\left[\hat{X}\right] &= \frac{\mathbb{E}\left[X_1\right] + \cdots + \mathbb{E}\left[X_n\right]}{n} = \overline{x} \end{align}

• and variance

$\sigma^2_\hat{X}:= \mathbb{E}\left[\left(\hat{X} - \overline{x}\right)^2\right] = \frac{\sigma^2 + \sigma^2 + \cdots + \sigma^2}{n^2} = \frac{\sigma^2}{n}$

### Central limit theorem continued

• More generally, if we are sampling from a population that has an unknown probability distribution, the sampling distribution of the sample mean will still be approximately Gaussian with mean $$\overline{x}$$ and variance $$\frac{\sigma^2}{n}$$ if the sample size $$n$$ is large.

• This is one of the most useful theorems in statistics, called the central limit theorem:

The central limit theorem (CLT)
Let $$X_1 , X_2 , \cdots , X_n$$ be a random sample of size $$n$$ taken from a population with mean $$\overline{x}$$ and finite variance $$\sigma^2$$ and $$\hat{X}$$ be the sample mean. Then the limiting form of the distribution of $Z = \frac{\hat{X} - \overline{x}}{\frac{\sigma}{\sqrt{n}}}$ as $$n \rightarrow \infty$$ is the standard normal distribution.
• Put another way, for $$n$$ sufficiently large, $$\hat{X}$$ has approximately a $$N\left(\overline{x}, \frac{\sigma^2}{n}\right)$$ distribution – this says the following.

• Suppose we take a sample of size $$n$$ and compute the sample mean $$\hat{x}$$.
• Then suppose we replicate this sample and record the observed realizations for the sample mean $$\hat{x}_1, \hat{x}_2, \cdots$$.
• If the sample size $$n$$ is large, these data points $$\hat{x}_1, \cdots$$ will be approximately bell shaped with the following properties:
• the bell will be centered approximately at $$\overline{x}$$, the true population mean;
• the spread of the data around the center will be given by approximately by the standard deviation $$\frac{\sigma}{\sqrt{n}}$$.
• Particularly, if $$n$$ is very large, the observed sample means will tend to be very close to the center (the true mean).

### Central limit theorem continued

• As a visualization of the concept, suppose that we have a random sample indexed by $$j$$ $X_{1,j}, \cdots, X_{n,j},$ where $$j$$ refers to the replication number.
• We will make replications for $$j=1,\cdots,m$$ and get a RV for sample mean indexed by $$j$$, $\hat{X}_j = \frac{1}{n}\sum_{i=1}^n X_{i,j}.$
• When we observe a realization of $$\hat{X}_j=\hat{x}_j$$ or respectively the sample $X_{1,j}=x_{1,j}, \cdots, X_{n,j}=x_{n,j},$ we record these fixed numerical values.