$(20$ pts$)$ One nice property for Gaussian distributions is that if we marginalize over any subset

of dimensions of a multivariate Gaussian, the resulting distribution is also Gaussian. To marginalize a

multivariate random variable means to find the distribution of only the marginalized random variables and

$$eliminate$$ from the distribution the other random variables. This is achieved by adding or integrating the

distribution function with respect to all other random variables, e$.$g$.$ for a continuous multivariate random

variable $(,)$ with PDF $,(,)$ we can marginalize on x by doing $()=,(,)$

$+\backslash$infty

$\backslash$infty $.$ This means

the distribution of a marginalized multivariate Gaussian distribution looks like the familiar $$bell shape$$

characteristic of a univariate Gaussian distribution.

a$)$ Use the below code to generate the PDF for a bivariate Gaussian. Then, in a separate figure, plot the

PDF when you marginalize $($integrate with respect to $),$ and overlay a plot of the PDF when you

marginalize $($integrate with respect to $),$ both on the same plot using legends.

b$)$ Using your knowledge of the Gaussian PDF$,$ estimate the mean of each marginal distribution and display

these means on your plot.

Show your plot and code.

import numpy as np

import matplotlib.pyplot as plt

from scipy.stats import multivariate$\_$normal

# Define mean $($mu$)$ and covariance matrix $($sigma$)$

mu $=$ np$.$array$([0,3])$

sigma $=$ np$.$array$([[5,-2],[-2,2]])$

# Create a grid of x$1$ and x$2$ values

x$1=$ np$.$arange$(-10,10\mathrm{.}1,0\mathrm{.}1)$

x$2=$ np$.$arange$(-10,10\mathrm{.}1,0\mathrm{.}1)$

X$1,$ X$2=$ np$.$meshgrid$($x$1,$ x$2)$

# Create a multivariate normal distribution

mvn $=$ multivariate$\_$normal$($mean$=$mu$,$ cov$=$sigma$)$

# Calculate the PDF values for each point in the grid

pdf$\_$values $=$ mvn$.$pdf$($np$.$column$\_$stack$(($X$1.$ravel$(),$ X$2.$ravel$())))$

# Reshape the PDF values to match the grid shape

F $=$ pdf$\_$values.reshape$($X$1.$shape$)$

# Create a contour plot

plt$.$contour$($x$1,$ x$2,$ F$)$

# Set plot attributes

plt$.$grid$($True$)$

plt$.$axis$(\text{'}$square$\text{'})$

plt$.$title$(\text{'}$Your Name CMPE $677,$ Hwk $1,$ Problem $10\text{'},$ fontsize$=12)$

# Save the plot as a PNG file

plt$.$savefig$(\text{'}$cmpe$677\_$hwk$1\_10.$png$\text{'},$ format$=\text{'}$png$\text{'})$

# Show the plot

plt$.$show$()$