## Normally Distributed

This most common distribution is your first step to the understanding of statistics.

 The normally distributed statistical process, also known as the Gaussian process exhibits normal or the Gaussian distribution - one of the basic continuous probability distributions. Most real world stochastic phenomena (whose mean and variance can be computed) exhibit a behavior that can be characterized by a normally distributed variable.

## Probability Density Function:

The probability density function of a normally distributed process is known as the Gaussian function, which appears in a characteristic bell shaped curve when plotted against the values of the variable.

Mathematically Where is the mean and 2 is the variance of the distribution.

The variance describes the spread of the distribution about the mean value. A larger 2 means greater scattering in the values taken by the variable. Resultantly, the probability of the variable to take the mean value is lower if the distribution has a smaller 2, which means that the variable is closely clustered around the mean value.

## Characteristic Function Curve of the Normally Distributed Process:

A graph of the probability density function (PDF) for normally distributed stochastic process with different values of standard deviation (square root of variance) is given below. As the variance or standard deviation increases, the height of the characteristic bell shaped curve (the probability of the variable to cluster closely around the mean value) decreases. A normal distribution also exhibits a normally distributed histogram, provided sample size is very large. ## Properties of a Normally Distributed Statistical Process:

• The PDF of the normally distributed process is symmetric about the mean.

• The mean divides the curve area into half.

• A Normally distributed process is completely described with two parameters: the mean and the variance.

## How to conduct the Process?

1. First of all, record data for the total weight of 8 random persons: 2. Then calculate mean value 3. Calculate variance. 4. Calculate standard deviation. 5. Determine probability density function. 6. Plot f(x) versus x over range of observations (452 ~ 692)

## Other Important Statistical Distributions:

Some other fundamental statistical probability distributions are:

• Uniform Distribution: The variable has equal probability of assuming any value over the range of possible values. As opposed to the normally distributed probability function which is bell shaped, uniform distribution probability density function is a horizontal line. Below graph shows the probability density function versus values of the variable. • Lognormal Distribution: The Log-normal distribution is the distribution of a random variable whose logarithm is normally distributed.

• t Distribution: A t distribution is achieved while estimating the mean of a normally distributed population when the sample size is small.

• Gamma Distribution: It is a family of distributions that vary greatly in characteristics depending upon its scale parameter (which describes the spread of the distribution) and its shape parameter (which describes the characteristic shape of the distribution).

## Summary

 Normally distributed statistical processes display one of the fundamental statistical probability distributions i.e. the normal distribution. The distribution can be completely described with just two parameters: the mean, and the variance. The graph of probability density function of normal distribution is a characteristic bell curve, which is symmetric about the mean. Other important probability distributions include uniform distribution, lognormal distribution, t distribution and gamma distribution.

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