MATH 204 Introduction to Statistics

class: center, middle, inverse, title-slide

.title[
# MATH 204 Introduction to Statistics
]
.subtitle[
## Lecture 8: Introduction to Random Variables
]
.author[
### JMG
]

---

## Goals for Lecture

* Introduce basic concepts and terminology for random variables. Textbook section 3.4

* We begin our discussion on the imporant notion of **random variable**.
    
--

* In this lecture, we focus on getting an intuitive grasp on the notion of random variable. In our next lecture, we will be more precise in our discussion of the topic.

---

## Introduction to Random Variables

- It's often useful to model a process using what's called a **random variable**.

- Suppose we toss a coin ten times and add up the number of heads that have appeared. Tossing a coin ten times is a random process, the total number of heads after ten tosses is a random variable.

- In general, a random variable assigns a **numerical value** to events from a random process.

- We will see later that random variables have distributions associated with them, and we want to be able to describe the distributions of random variables.

- The sample mean is an important example of a random variable and its distribution is an example of a **sampling distribution**.

---

## Notation for Random Variables

- We typically denote random variables by capital letters at the end of the alphabet such as `\(X\)`, `\(Y\)`, or `\(Z\)`.

- For example, let `\(X\)` be the random variable that is the sum of the number of heads that we obtain after tossing a coin ten times. The possible values that `\(X\)` can take on are `\(X=0\)`, `\(X=1\)`, `\(X=2\)`, `\(\ldots\)`, `\(X=10\)`.

- Typical questions that we ask are ones such as, what is the probability that `\(X=2\)`, or what is the probability that `\(X\)` is less than 5. What do these questions mean in the context of our coin tossing example?

- The distribution of `\(X\)` allows us to answer such questions.

---

## A Look at Some Data

- Let's look at six rounds of tossing a coin ten times. At each toss, we record 1 if we land heads and 0 if we land tails. Then we can add up the 1's to get the total number of heads after ten tosses. Our data might look as follows:

<table class="table" style="margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> X1 </th>
   <th style="text-align:right;"> X2 </th>
   <th style="text-align:right;"> X3 </th>
   <th style="text-align:right;"> X4 </th>
   <th style="text-align:right;"> X5 </th>
   <th style="text-align:right;"> X6 </th>
   <th style="text-align:right;"> X7 </th>
   <th style="text-align:right;"> X8 </th>
   <th style="text-align:right;"> X9 </th>
   <th style="text-align:right;"> X10 </th>
   <th style="text-align:right;"> total_heads </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> round_1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 5 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> round_2 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 4 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> round_3 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 7 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> round_4 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 4 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> round_5 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 4 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> round_6 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 4 </td>
  </tr>
</tbody>
</table>

- Let's repeat this process many more times and create a barplot that shows the number of times each of the values 0, 1, ..., 10 occurs.

---

## The Distribution of the Number of Heads

- If we divide all of the counts by the total, we get the density. This provides an estimate for the probability value of each outcome.

---

## Probabilities for Number of Heads

This plot shows the density instead of count for the previous barplot.

- From this plot, we can easily estimate probability values. For example, what is the probability of getting 4 heads out of ten tosses? It's about 0.2. What is the probability of getting less than three heads? It's about 0.04 + 0.01 + 0.0025 = 0.0525.

---

# The Mean and Variance for Number of Heads

We can compute the mean and variance for the number of heads, in which case we get:

```
## [1] "The mean is 5.035400"
```

```
## [1] "The variance is 2.385024"
```

- This tells us that the "average" number of heads out of ten tosses is about 5. How does this correspond with your real life experience or expectations?

- These values provide estimates for the **expected value** and **variance** of our random variable. These concepts will be defined and discussed in detail in the next lecture.

---

## Summary

In this lecture, we introduced the basic notion of random variable.

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## Notes

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## Notes

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## Notes