Some basic stuff for Class 2

Random numbers and PDFs

Tom Solomon, February 2021

import numpy as np
from scipy import stats

import matplotlib as mpl
import matplotlib.pyplot as plt

Get a random integer and a random floating point number

low = 3
high = 7
np.random.randint(low,high+1) #Random integer between 3 and 7

3

rfloat = np.random.random_sample() #Random number between 0 and 1
rfloat

0.2190439346827222

Now let's look at distributions. First, uniform, then Gaussian, then Poisson

xvals = np.linspace(0.,10.,200)
p_uniform = np.array([0.1 for i in range(200)]) #This is boring -- just 0.1 for every value

plt.figure()
plt.plot(xvals,p_uniform)

[<matplotlib.lines.Line2D at 0x1f9d67d00d0>]

Now for the Gaussian (normal distribution). $p(x) = \frac{1}{\sigma \sqrt{2\pi}}exp\left[-\frac{(x-\bar{x})^2}{2\sigma^2}\right]$

meanval = 5.0
σ = 2.0
p_gau = (1/(σ*np.sqrt(2.0*np.pi)))*np.exp(-1.0*(xvals-meanval)**2/(2.0*σ**2))
plt.figure()
plt.plot(xvals,p_gau)

[<matplotlib.lines.Line2D at 0x1f9d6afa790>]

Now for Poisson: $p(N) = \frac{exp(-a)a^N}{N!}$ where $a$ is the mean value of $N$.

meanval = 5
xintvals = np.array([i for i in range(11)])  # Up to 10
p_poi = np.array([np.exp(-1*meanval)*meanval**i/np.math.factorial(i) for i in range(11)])
plt.figure()
plt.scatter(xintvals,p_poi)

<matplotlib.collections.PathCollection at 0x1f9d6b5fac0>

Let's try plotting the previous two graphs together

plt.figure()
plt.scatter(xintvals,p_poi)
plt.plot(xvals,p_gau)

[<matplotlib.lines.Line2D at 0x1f9d6bcc910>]