# Homework 11¶

• Type your name and email in the “Student details” section below.

• Develop the code and generate the figures you need to solve the problems using this notebook.

• For the answers that require a mathematical proof or derivation you can either:

• Type the answer using the built-in latex capabilities. In this case, simply export the notebook as a pdf and upload it on gradescope; or

• You can print the notebook (after you are done with all the code), write your answers by hand, scan, turn your response to a single pdf, and upload on gradescope.

• The total homework points are 100. Please note that the problems are not weighed equally.

Note

• This is due before the beginning of the next lecture.

• Please match all the pages corresponding to each of the questions when you submit on gradescope.

## Student details¶

• First Name:

• Last Name:

• Email:

import matplotlib.pyplot as plt
%matplotlib inline
import seaborn as sns
sns.set(rc={"figure.dpi":100, 'savefig.dpi':300})
sns.set_context('notebook')
sns.set_style("ticks")
from IPython.display import set_matplotlib_formats
set_matplotlib_formats('retina', 'svg')
import numpy as np
import scipy.stats as st


## Problem 1 - Calculating expectations for discrete random variables¶

Consider the Categorical random variable $$X \sim \text{Categorical}(0.2, 0.4, 0.4)$$ taking three discrete values $$1, 2,$$ and $$3$$. Find the numerical answer for the following. Hint: You can do it by hand or write some code. It is up to you.

• $$\mathbf{E}[X] = ?$$

# Your code here

• $$\mathbf{E}[X^2] = ?$$

# Your code here

• $$\mathbf{V}[X] = ?$$

# Your code here

• $$\mathbf{E}[e^X] = ?$$

# Your code here


## Problem 2 - Calculate the expectation and variance of a continuous random variable¶

Take an exponential random variable, see Problem 2 of Homework 10, with rate parameter $$\lambda$$:

$T \sim \text{Exp}(\lambda).$

The PDF of the exponential is:

$p(t) = \lambda e^{-\lambda t},$

for $$t\ge 0$$ and zero otherwise.

• Prove mathematically that $$\mathbf{E}[T] = \lambda^{-1}$$. Hint: You need to do the integral $$\int_0^\infty tp(t)dt$$ using integration by parts.

• Prove mathematically that $$\mathbf{V}[T] = \lambda^{-2}$$. Hint: Use integration by parts to find $$\mathbf{E}[T^2]=\int_0^\infty t^2 p(t)dt$$ and then use one of the properties of the variance.

## Problem 3 - Properties of expectations¶

Let $$X$$ be a random variable with expectation $$\mathbf{E}[X] = 3$$ and variance $$\mathbf{V}[X] = 2$$. Calculate the following expressions.

• $$\mathbf{E}[4X] = ?$$

• $$\mathbf{E}[X + 2] = ?$$

• $$\mathbf{E}[3X + 1] = ?$$

• $$\mathbf{V}[5X] = ?$$

• $$\mathbf{V}[X + 3] = ?$$

• $$\mathbf{V}[2X + 1] = ?$$

• $$\mathbf{E}[X^2] = ?$$