---
title: "Homework assignment #1"
author: "Milica Cudina"
date: "`r Sys.Date()`"
output: pdf_document
---
<!-- The author of this template is Dr. Gordan Zitkovic.-->
<!-- The code chunk below contains some settings that will  -->
<!-- make your R code look better in the output pdf file.  -->
<!-- If you are curious about what each option below does, -->
<!-- go to https://yihui.org/knitr/options/ -->
<!-- If not, feel free to disregard everything ...  -->
```{r echo=FALSE, warning=FALSE, include=FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  fig.align="center",
  fig.pos="t",
  strip.white = TRUE
)
```
<!-- ... down to here. -->

---

# Definitions
## Problem 1. **(5 points)**
Write down the definition of *independence* of two *events*.

## Problem 2. **(5 points)**
Write down the definition of the *cumulative distribution function* of a random variable.


---

# Problems
## Problem 3. **(9 points)**
Let $\Omega = \{a_1,  a_2, a_3, a_4, a_5\}$ be an outcome space, and let $\mathbb{P}$ be a probability on $\Omega$. Assume that $\mathbb{P}[A] = 0.5$, $\mathbb{P}[B]=0.4$, $\mathbb{P}[C]=0.4$, and $\mathbb{P}[D]=0.2$, where
\begin{equation}\nonumber
 \begin{split}
 A &= \{a_1,a_2,a_3\}, \ B =
  \{a_2,a_3,a_4\},\\  C &= \{a_3,a_5\}\text{ and } D=\{a_4\}.
 \end{split}
\end{equation}
Are the events $A$ and $B$ independent? Why?


---

# Textbook problems
## Problem 4. **(4 points)** 
Solve **Problem 3.4** from the textbook. 


## Problem 5. **(4 points)** 
Solve **Problem 3.6** from the textbook. 


## Problem 6. **(a-e are 2 points each; f is 5 points=15 points total)**\
Solve **Problem 3.8** from the textbook.


## Problem 7. **(4 points)** 
Solve **Problem 3.20** from the textbook.


## Problem 8. **(4 points)** 
Solve **Problem 3.22** from the textbook.