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Normal Distribution

Definition

The normal pdf is

`$y=f\left(x|\mu ,\sigma \right)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{\frac{-{\left(x-\mu \right)}^{2}}{2{\sigma }^{2}}}.$`

Background

The normal distribution is a two-parameter family of curves. The first parameter, µ, is the mean. The second, σ, is the standard deviation. The standard normal distribution (written Φ(x)) sets µ to 0 and σ to 1.

Φ(x) is functionally related to the error function, erf.

`$erf\left(x\right)=2\Phi \left(x\sqrt{2}\right)-1$`

The first use of the normal distribution was as a continuous approximation to the binomial.

The usual justification for using the normal distribution for modeling is the Central Limit Theorem, which states (roughly) that the sum of independent samples from any distribution with finite mean and variance converges to the normal distribution as the sample size goes to infinity.

Parameters

To use statistical parameters such as mean and standard deviation reliably, you need to have a good estimator for them. The maximum likelihood estimates (MLEs) provide one such estimator. However, an MLE might be biased, which means that its expected value of the parameter might not equal the parameter being estimated. For example, an MLE is biased for estimating the variance of a normal distribution. An unbiased estimator that is commonly used to estimate the parameters of the normal distribution is the minimum variance unbiased estimator (MVUE). The MVUE has the minimum variance of all unbiased estimators of a parameter.

The MVUEs of parameters µ and σ2 for the normal distribution are the sample mean and variance. The sample mean is also the MLE for µ. The following are two common formulas for the variance.

 ${s}^{2}=\frac{1}{n}\sum _{i=1}^{n}{\left({x}_{i}-\overline{x}\right)}^{2}$ (1)
 ${s}^{2}=\frac{1}{n-1}\sum _{i=1}^{n}{\left({x}_{i}-\overline{x}\right)}^{2}$ (2)

where

`$\overline{x}=\sum _{i=1}^{n}\frac{{x}_{i}}{n}$`

Equation 1 is the maximum likelihood estimator for σ2, and equation 2 is the MVUE.

To fit the normal distribution to data and find the parameter estimates, use `normfit`, `fitdist`, or `mle`.

• For uncensored data, `normfit` and `fitdist` find the unbiased estimates of the distribution parameters, and `mle` finds the maximum likelihood estimates.

• For censored data, `normfit`, `fitdist`, and `mle` find the maximum likelihood estimates.

Unlike `normfit` and `mle`, which return parameter estimates, `fitdist` returns the fitted probability distribution object `NormalDistribution`. The object properties `mu` and `sigma` of store the parameter estimates.

As an example, suppose you want to estimate the mean, µ, and the variance, σ2, of the heights of all fourth grade children in the United States. The function `normfit` returns the MVUE for µ, the square root of the MVUE for σ2, and confidence intervals for µ and σ2. Here is a playful example modeling the heights in inches of a randomly chosen fourth grade class.

```rng default; % For reproducibility height = normrnd(50,2,30,1); % Simulate heights [mu,s,muci,sci] = normfit(height)```
```mu = 51.1038 ```
```s = 2.6001 ```
```muci = 2×1 50.1329 52.0747 ```
```sci = 2×1 2.0707 3.4954 ```

Note that `s^2` is the MVUE of the variance.

`s^2`
```ans = 6.7605 ```

Examples

Compute and Plot the Normal Distribution pdf

Compute the pdf of a standard normal distribution, with parameters equal to 0 and equal to 1.

```x = [-3:.1:3]; norm = normpdf(x,0,1);```

Plot the pdf.

```figure; plot(x,norm)```