Normal distribution is most common in real life scenarios be it modeling any reliability performance parameter at a specific time. Central Limit Theorem (CLT) shows why normal distribution occurs so often.

## Central Limit Theorem (CLT):

Definition: Distribution of the sum (or average) of a large number of independent, identically distributed random variables, regardless of their original distribution, will tend to normal (bell-shaped or Gaussian) distribution, given the sample size is sufficiently large.

Usage: CLT enables to use normal distribution to make inference and recommendation about the population mean, even if the population data is not normally distributed. Thus, we can apply hypothesis testing and confidence intervals for many applications.

## Normal Distribution:

Definition: Continuous probability distribution having characteristic symmetric, bell-shape curve. Described by mean (µ) and standard deviation (s).

Some important properties of normal distribution are:

- Mean, median and mode are all equal.
- 68-95-99.7 Rule: 68% of the data lies within one standard deviation of the mean, 95% within two standard deviations and 99.7% within three standard deviations.
- Pdf and cdf are represented by the following equations:

pdf given by

$$ \displaystyle f\left(t\right)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}} $$cdf given by

$$ \displaystyle F\left(t\right)=\frac{1}{2}\left[1+erf\left(\frac{x-\mu}{\sigma\sqrt{2}}\right)\right] $$Where key parameters µ = mean and s = standard deviation

CLT and Normal Distribution

CLT is the foundational basis as to why normal distribution is so common in statistics and real-life scenarios.

- Sampling Distribution of Sample Mean: The sampling distribution of the sample means will be normally distributed, even if the original population is not normal. Very useful to draw inferences about the population.
- We can use normal distribution in constructing confidence intervals and hypothesis testing about population means when sample size is large.
- For smaller sample size, if population is normal or close to normal, sample means will still follow normal distribution.

There are many practical software packages such as JMP and Python numpy, matplotlib libraries for modeling CLT and normal distribution.

Source: Reliawiki

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