# Statistical Distribution Functions

This post is mainly about observing the theoretical distributions with variation of the terms that make the equations. There will be a lot of animations which will hopefully help improve your understanding of these concepts.

1. Normal Distribution
2. Exponential Distribution
3. Gamma Distribution
4. Weibull Distribution

One of the most commonly seen distributions and one of the first ones taught in probability courses. We see this normal everywhere. Let us start with the expression.

$\frac{e^{-\frac{(x-\mu )^2}{2 \sigma ^2}}}{\sqrt{2 \pi } \sigma }$

We will now change the quantities one by one and see the variation in the graph. The one below is an example graph done by using $\sigma$ = 1 and $\mu$ = 1.

.

The graph below is the one with changing values of $\mu$ or mean.

.

The graph below is the one with changing values of $\sigma$ or standard deviation. Standard deviation tells about the variability of a particular normal distribution. We can observe that the as the $\sigma$ is increasing, the graph becomes shorter and wider. Also note that the $\sigma$ does not change the position of the peak or the mean.

The exponential distribution can be written as $\lambda$ $e^{\lambda x}$

In the case of this distribution, the parameter of interest is $\lambda$ and the mean value of the distribution is $\frac{1}{\lambda }$. The graph below is plotted for $\lambda$ = 1.

This graph below will show what happens to the distribution with various values of $\lambda$

The gamma distribution can be expressed as  $\frac{\beta ^{\alpha }}{\Gamma (\alpha)}$  $x^{\alpha -1} e^{-\beta x}$ where $\alpha$ and $\beta$ is the random variables and $x$ is the random variable.

Let us look at some plots of the variable by changing $\alpha$ and $\beta$ independently.

$\beta$ = 1; various values of $\alpha$

$\beta$ ranging from 0.01 to 1 in steps of 0.01 for $\alpha$ values of 1 to 10.

Weibull Distribution is denoted by $\frac{\alpha}{\beta^\alpha} x^{\alpha-1} e^{{-(\frac{x}{\beta}})^{\alpha}}$ where $x$ is the random variable and  $\alpha$ and $\beta$ are the parameters. Let us look at the behavior of the distribution with various values of $\alpha$ and $\beta$.

$\alpha$ varying from 1 to 10 and $\beta$ =1

Weibull distribution with $\alpha$  from 1 to 10 and $\beta$ changing from 1 to 10

The lognormal distribution graphs will be added soon.

.