Physics 310

Class 2

Tuesday January 28, 2025

Goals:

• Introduce tools and functions for plotting
• Introduce tools for generating random numbers and histograms
• Experiment with PDFs (probability distribution functions)

## Import relevant packages
import numpy as np
import matplotlib.pyplot as plt

from scipy import stats

The linspace function and optional arguments.

Often, we will fine that we need to generate a regularly spaced array of numbers, such as when creating an x-axis, or a set of independent variables

The np.linspace is really valuable in this regard. What does it do? Look up its syntax by running the following command

?np.linspace

Run the following commands np.linspace(0,30) and note what each varible does

From the help file, identify required and optional arguments. Note that the number of samples returned is an optional argument. What is its default value?

Below, we will demonstrate a few methods for generating random numbers. Importantly, the random numbers are drawn from a given statistical distribution. We consider the Uniform, Normal (Gaussian), Binomial, and Poisson distributions.

The Uniform Distribution

# Sample n random numbers in interval [0.0,1.0]:
n = 10
stats.uniform.rvs(size=n)        # "uniform" in the command indicates all values are equally likely

array([0.96428521, 0.22810387, 0.46150645, 0.02136174, 0.60182643,
       0.80169691, 0.70136798, 0.04686054, 0.81178822, 0.08349542])

• How can you graphically demonstrate that the numbers drawn are truly uniform?

#?np.random.random_sample

Normal Distribution

This is given by

$p(x) = \frac{1}{\sigma \sqrt{2\pi}}exp\left[-\frac{(x-\bar{x})^2}{2\sigma^2}\right]$

What do the symbols $\sigma$, and $\bar{x}$ represent. The snippet of code below generates 10 samples from a normal distribution

# Sampling from normal distribution
n = 10
mean = 10.
sigma = 2.
stats.norm.rvs(mean, sigma, size=n)

# Sampling from normal distribution
n = 1000
mean = 10.
sigma = 2.
ynorm = stats.norm.rvs(mean, sigma, size=n)  

x = np.linspace(0,20,200)  # make an array of 200 evenly spaced values from 0 to 20
y = stats.norm.pdf(x, mean, sigma)      # determine the value of the pdf at each of the points in 'x'
plt.title("pdf of normal distribution")
plt.xlabel("$x$")
plt.ylabel("$p(x)$")
plt.grid()
plt.plot(x, y);

Consider the following snippet of code

n = 1000
mean = 10.
sigma = 2.
ynorm = stats.norm.rvs(mean, sigma, size=n)

Before running it, describe what it will do.

Now generate a histogram of using:

plt.hist(ynorm)
plt.xlabel("value")
plt.ylabel("occurences")
plt.title("Histogram; equal sized bins")

Now the plt.hist function has a number of optional variables. For example, you can normalize it using density = True, and you can change the relative width of the cells by using rwidth = ...

Try this snippet of code, and think about what each line an each optional variable does

plt.hist(ynorm, density = True,rwidth=0.8,label = 'samples')
plt.plot(x, y, label = 'pdf');
plt.legend()

Label the x - axis "value" and the y axis "occurrences"

Binomial Distribution

The stats module implements dozens of other distributions. For example, to generate random values from a binomial distribution, you may use

stats.binom.rvs(n,p,size = 100) where n and p are the two required variables for the binomial distribution.

• Make a plot of the Probability Mass Function for a binom distribution where p = 0.4, n = 5. Your axes should be "Probabability of n successes" versus "n" on the x axis.

Poisson Distribution

stats.poisson.rvs(mean, size=100) generates a sample of 100 random variables form the poisson distribution. Note that the poisson distribution is a function of a single variable. Namely the mean number of counts.

Experiment with generating random numbers and histograms of these distributions.

• Make a histogram of random numbers which are poisson distributed for the cases:

$\mu = 1$

$\mu = 4$

$\mu = 12$