Homework 11
Contents
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.
\(\mathbf{E}[X] = ?\)
Answer:
# Your code here
\(\mathbf{E}[X^2] = ?\)
Answer:
# Your code here
\(\mathbf{V}[X] = ?\)
Answer:
# Your code here
\(\mathbf{E}[e^X] = ?\)
Answer:
# 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 \(\lambda\):
The PDF of the exponential is:
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.
Answer:
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.
Answer:
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.
\(\mathbf{E}[4X] = ?\)
Answer:
\(\mathbf{E}[X + 2] = ?\)
Answer:
\(\mathbf{E}[3X + 1] = ?\)
Answer:
\(\mathbf{V}[5X] = ?\)
Answer:
\(\mathbf{V}[X + 3] = ?\)
Answer:
\(\mathbf{V}[2X + 1] = ?\)
Answer:
\(\mathbf{E}[X^2] = ?\)
Answer: