Section 2.3 — Inventory of discrete distributions

This notebook contains all the code examples from Section 2.3 Inventory of discrete distributions of the No Bullshit Guide to Statistics.

Notebook setup

# load Python modules
import numpy as np
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt

# Figures setup
sns.set_theme(
    context="paper",
    style="whitegrid",
    palette="colorblind",
    rc={'figure.figsize': (6,4)},
)

%config InlineBackend.figure_format = 'retina'

# set random seed for repeatability
np.random.seed(42)

%pip install -q ministats

Note: you may need to restart the kernel to use updated packages.

from ministats import plot_pmf
from ministats import plot_cdf

Definitions

Math prerequisites

Combinatorics

See SciPy docs:

• https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.factorial.html

• https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.perm.html

• https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.comb.html

Factorial

from scipy.special import factorial
# ALT.
# from math import factorial

factorial(4)

24.0

factorial(1), factorial(2), factorial(3)

(1.0, 2.0, 6.0)

[factorial(k) for k in [5,6,7,8,9,10,11,12]]

[120.0, 720.0, 5040.0, 40320.0, 362880.0, 3628800.0, 39916800.0, 479001600.0]

factorial(15)

1307674368000.0

import numpy as np
np.log(factorial(15))/np.log(10)

12.116499611123398

Permutations

from scipy.special import perm

perm(5,2)

20.0

perm(5,1), perm(5,2), perm(5,3), perm(5,4), perm(5,5)

(5.0, 20.0, 60.0, 120.0, 120.0)

Combinations

from scipy.special import comb

comb(5,2)

10.0

Discrete distributions reference

Discrete uniform

# import the model family
from scipy.stats import randint

# choose parameters
alpha = 1  # start at
beta = 4   # stop at

# create the rv object
rvU = randint(alpha, beta+1)

# use one of the rv object's methods

The limits of the sample space of the random variable rvU can be obtained by calling its .support() method.

rvU.support()

(1, 4)

rvU.mean()

2.5

rvU.var()

1.25

rvU.std()  # = np.sqrt(rvU.var())

1.118033988749895

Probability mass function

for x in range(1,4+1):
    print(f"f_U({x})  = ", rvU.pmf(x))

f_U(1)  =  0.25
f_U(2)  =  0.25
f_U(3)  =  0.25
f_U(4)  =  0.25

To create a stem-plot of the probability mass function \(f_U\), we can use the following three-step procedure:

1. Create a range of inputs xs for the plot.

2. Compute the value of \(f_U =\) rvU for each of the inputs and store the results as list of values fUs.

3. Plot the values fUs by calling the function plt.stem(xs,fUs).

import numpy as np
import matplotlib.pyplot as plt

xs = np.arange(0,8+1)
fUs = rvU.pmf(xs)
plt.stem(xs, fUs, basefmt=" ");

# ALT
plot_pmf(rvU, xlims=[0,8+1]);

Cumulative distribution function

for b in range(1,4+1):
    print(f"F_U({b}) =", rvU.cdf(b))

F_U(1) = 0.25
F_U(2) = 0.5
F_U(3) = 0.75
F_U(4) = 1.0

import numpy as np
import seaborn as sns

xs = np.linspace(0,8,1000)
FUs = rvU.cdf(xs)
sns.lineplot(x=xs, y=FUs);

# ALT
plot_cdf(rvU, xlims=[0,8], rv_name="U");

Let’s generate 10 random observations from random variable rvU:

rvU.rvs(10)

array([3, 4, 1, 3, 3, 4, 1, 1, 3, 2])

Bernoulli

from scipy.stats import bernoulli

rvB = bernoulli(p=0.3)

rvB.support()

(0, 1)

rvB.mean(), rvB.var()

(0.3, 0.21)

rvB.rvs(10)

array([0, 0, 1, 0, 1, 0, 1, 1, 0, 0])

plot_pmf(rvB, xlims=[0,5]);

Binomial

We’ll use the name rvX because rvB was already used for the Bernoulli random variable above.

from scipy.stats import binom

n = 20
p = 0.14
rvX = binom(n,p)

rvX.support()

(0, 20)

rvX.mean(), rvX.var()

(2.8000000000000003, 2.4080000000000004)

plot_pmf(rvX, xlims=[0,30]);

Poisson

from scipy.stats import poisson
lam = 10
rvP = poisson(lam)

rvP.pmf(8)

0.11259903214902009

rvP.cdf(8)

0.3328196787507191

## ALT. way to compute the value F_P(8) =
# sum([rvP.pmf(x) for x in range(0,8+1)])

plot_pmf(rvP, xlims=[0,30]);

Geometric

from scipy.stats import geom

rvG = geom(p = 0.2)

rvG.support()

(1, inf)

rvG.mean(), rvG.var()

(5.0, 20.0)

plot_pmf(rvG, xlims=[0,40]);

Negative binomial

from scipy.stats import nbinom

r = 10
p = 0.5
rvN = nbinom(r,p)

rvN.support()

(0, inf)

rvN.mean(), rvN.var()

(10.0, 20.0)

plot_pmf(rvN, xlims=[0,40]);

Hypergeometric (optional)

from scipy.stats import hypergeom

a = 30   # number of success balls
b = 40   # number of failure balls
n = 20   # how many we're drawing

rvH = hypergeom(a+b, a, n)

rvH.support()

(0, 20)

rvH.mean(), rvH.var()

(8.571428571428571, 3.54924578527063)

plot_pmf(rvH, xlims=[0,30]);

Tomatoes salad probabilities

a = 3   # number of good tomatoes
b = 4   # number of rotten tomatoes
n = 2   # how many we're drawing

rvHe = hypergeom(a+b, a, n)


plot_pmf(rvHe, xlims=[0,3])

rvHe.pmf(0), rvHe.pmf(1), rvHe.pmf(2)

(0.28571428571428575, 0.5714285714285715, 0.14285714285714288)

Number of dogs seen by Amy

a = 7        # number dogs
b = 20 - 7   # number of other animals
n = 12       # how many "patients" Amy will see today

rvD = hypergeom(a+b, a, n)

# Pr of exactly five dogs
rvD.pmf(5)

0.2860681114551084

plot_pmf(rvD, xlims=[0,10]);

Modelling real-world data using probability

TODO: add simple inference and plots

Review of computer models

Discussion