## Introduction

To generate random numbers from a normal (Gaussian) distribution in Python, you can use the random module or the `numpy`

library. Below are examples demonstrating both methods.

## Utilizing the numpy random module

The random module in numpy is particularly useful when working with large datasets, as it allows you to generate random numbers and arrays quickly and accurately for testing purposes.

### Generate random numbers from a standard normal distribution

To generate random numbers from a standard normal distribution with a mean of 0 and a standard deviation of 1, use the following method:

`import numpy as np`

`import matplotlib.pyplot as plt`

`data = np.random.randn(100000)`

`hx, hy, _ = plt.hist(data, bins=50, density=1,color="lightblue")`

`plt.ylim(0.0,max(hx)+0.05)`

`plt.title('Generate random numbers \n from a standard normal distribution with python')`

`plt.grid()`

`plt.savefig("numpy_random_numbers_stantard_normal_distribution.png", bbox_inches='tight')`

`plt.show()`

The above code will generate 100,000 random numbers from a standard normal distribution with a mean of 0 and a standard deviation of 1. The resulting histogram plot shows the bell-shaped curve that is characteristic of a normal distribution.

### Generate random numbers from a normal distribution

Generating random numbers from a standard normal distribution equips us with the ability to generate random numbers from any normal distribution. This can be accomplished using the formula $$X = Z * \sigma + \mu$$ where Z represents random numbers from a standard normal distribution, $\sigma$ denotes the standard deviation, and $\mu$ signifies the mean.

`import numpy as np`

`import matplotlib.pyplot as plt`

`mu = 10.0`

`sigma = 2.0`

`data = np.random.randn(100000) * sigma + mu`

`hx, hy, _ = plt.hist(data, bins=50, density=1,color="lightblue")`

`plt.ylim(0.0,max(hx)+0.05)`

`plt.title('Generate random numbers \n from a normal distribution with python')`

`plt.grid()`

`plt.savefig("numpy_random_numbers_normal_distribution.png", bbox_inches='tight')`

`plt.show()`

The above code will generate 100,000 random numbers from a normal distribution with a mean of 10 and a standard deviation of 2. The resulting histogram plot shows that the data is more concentrated around the mean compared to the previous example which had a standard deviation of 1.

## Using the gauss function provided by the Python random module

An alternative approach is to utilize the gauss function provided by the Python random module

`from random import gauss`

`gauss(100,15)`

For instance, here is a randomly generated number as an example:

`82.13399330852374`

It is then possible to create a list of random numbers following a Gaussian distribution like this:

`x = [gauss(100,15) for i in range(10000)]`

and plot it using matplotlib:

`from random import gauss`

`import matplotlib.pyplot as plt`

`x = [gauss(100,15) for i in range(10000)]`

`num_bins = 50`

`n, bins, patches = plt.hist(x, num_bins, density=1, facecolor='green', alpha=0.5)`

`plt.show()`

Additionally, the output of this code can be saved as an image or further manipulated using matplotlib's extensive features:

This allows for further analysis and visualization of data following a Gaussian distribution. Other distributions such as Poisson, Binomial, and Exponential can also be simulated and plotted in a similar manner.

## References

Links | Site |
---|---|

Normal distribution | en.wikipedia.org |

numpy.random.randn | docs.scipy.org |

numpy.random.normal | docs.scipy.org |