Section 2.5 — Multiple continuous random variables

This notebook contains all the code examples from Section 2.5 Multiple continuous random variables of the No Bullshit Guide to Statistics.

Notebook setup

We’ll start by importing the Python modules we’ll need for this notebook.

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

from plot_helpers import RCPARAMS
RCPARAMS.update({"figure.figsize": (7, 2)})
sns.set_theme(
    context="paper",
    style="whitegrid",
    palette="colorblind",
    rc=RCPARAMS,
)
%config InlineBackend.figure_format = 'retina'

%pip install -q ministats

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

Definitions

Random variables

• random variable \(X\): a quantity that can take on different values.

• sample space \(\mathcal{X}\): describes the set of all possible outcomes of the random variable \(X\).

• outcome: a particular value \(\{X = x\}\) that can occur as a result of observing the random variable \(X\).

• event subset of the sample space \(\{a \leq X \leq b\}\) that can occur as a result of observing the random variable \(X\).

• \(f_X\): the probability density function (PDF) is a function that assigns probabilities to the different outcome in the sample space of a random variable. The probability distribution function of the random variable \(X\) is a function of the form \(f_X: \mathcal{X} \to \mathbb{R}\).

Multiple random variables

Example 3: bivariate normal distribution

TODO: formula for general bivarate normal

from scipy.stats import multivariate_normal

# parameters
mu = [10, 5]
Sigma = [[  3**2,     0.75*3*1],
         [  0.75*3*1,     1**2]]


# multivariate normal
rvXY = multivariate_normal(mu, Sigma)

rvXY.pdf((10,5))

0.08020655225672235

from scipy.integrate import dblquad

def fXY(x,y):
    """
    Adapter function because `dblquad` expects the function
    we're integrating to be of the form f(y,x), and not f(x,y).
    """
    return rvXY.pdf([y,x])

dblquad(fXY, a=11, b=np.inf, gfun=6, hfun=np.inf)[0]

0.1372330649420418

1 - rvXY.cdf((11,np.inf)) - rvXY.cdf((np.inf,6)) + rvXY.cdf((11,6))

0.13723306482268627

Joint probability density functions

TODO: formulas

from ministats import plot_joint_pdf_contourf

xlims = [3, 17]
ylims = [1.5, 8.5]

plot_joint_pdf_contourf(rvXY, xlims=xlims, ylims=ylims);

from ministats import plot_joint_pdf_surface

viewdict = dict(elev=60., azim=-110, roll=-16)
plot_joint_pdf_surface(rvXY, xlims=xlims, ylims=ylims, viewdict=viewdict);

Marginal density functions

TODO: formulas

from ministats.book.figures import plot_joint_pdf_and_marginals

plot_joint_pdf_and_marginals(rvXY, xlims=xlims, ylims=ylims);

---------------------------------------------------------------------------
ModuleNotFoundError                       Traceback (most recent call last)
Cell In[10], line 1
----> 1 from ministats.book.figures import plot_joint_pdf_and_marginals
      3 plot_joint_pdf_and_marginals(rvXY, xlims=xlims, ylims=ylims);

ModuleNotFoundError: No module named 'ministats.book'

Conditional probability density functions

from ministats.book.figures import plot_slices_through_joint_pdf

xcuts = range(2, 15, 1)
plot_slices_through_joint_pdf(rvXY, xlims=[3,17], ylims=[0,10], xcuts=xcuts);

from ministats.book.figures import plot_conditional_fYgivenX

xcuts = range(6, 15, 1)
plot_conditional_fYgivenX(rvXY, xlims=[3,17], ylims=[0,10], xcuts=xcuts);

Examples

Example ?: Multivariate normal

# TODO

Useful probability formulas

Multivariable expectation

Mean

rvXY.mean

array([10.,  5.])

Covariance

Sigma = rvXY.cov
Sigma

array([[9.  , 2.25],
       [2.25, 1.  ]])

covXY = Sigma[0,1]
covXY

2.25

Correlation

stdX = np.sqrt(Sigma[0,0])
stdY = np.sqrt(Sigma[1,1])

corrXY = covXY / (stdX * stdY)
corrXY

0.75

Independent, identically distributed random variabls

TODO formulas

Discussion

Exercises

Links