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.

In [1]:

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

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

In [2]:

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'

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'

In [3]:

%pip install -q ministats

%pip install -q ministats

[notice] A new release of pip is available: 24.3.1 -> 25.1.1
[notice] To update, run: pip install --upgrade pip
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

In [4]:

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)

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)

In [5]:

rvXY.pdf((10,5))

rvXY.pdf((10,5))

Out[5]:

0.08020655225672235

In [6]:

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]

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]

Out[6]:

0.1372330649420418

In [7]:

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

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

Out[7]:

0.13723306482268627

In [ ]:

Joint probability density functions¶

TODO: formulas

In [8]:

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_contourf xlims = [3, 17] ylims = [1.5, 8.5] plot_joint_pdf_contourf(rvXY, xlims=xlims, ylims=ylims);

In [9]:

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);

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

In [10]:

from ministats.book.figures import plot_joint_pdf_and_marginals

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

from ministats.book.figures import plot_joint_pdf_and_marginals plot_joint_pdf_and_marginals(rvXY, xlims=xlims, ylims=ylims);

Conditional probability density functions¶

In [11]:

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

In [12]:

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);

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¶

In [13]:

# TODO

# TODO

Useful probability formulas¶

Multivariable expectation¶

Mean¶

In [14]:

rvXY.mean

rvXY.mean

Out[14]:

array([10.,  5.])

Covariance¶

In [15]:

Sigma = rvXY.cov
Sigma

Sigma = rvXY.cov Sigma

Out[15]:

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

In [16]:

covXY = Sigma[0,1]
covXY

covXY = Sigma[0,1] covXY

Out[16]:

2.25

Correlation¶

In [17]:

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

corrXY = covXY / (stdX * stdY)
corrXY

stdX = np.sqrt(Sigma[0,0]) stdY = np.sqrt(Sigma[1,1]) corrXY = covXY / (stdX * stdY) corrXY

Out[17]:

0.75

Independent, identically distributed random variabls¶

TODO formulas

Discussion¶

Exercises¶

Links¶