ECON 3818

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# ECON 3818
## Expectations
### Kyle Butts
### 27 September 2021

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## Expectations

---
# Outline

.hi.coral[Expectation]

- Definition
- Properties

.hi.ruby[Variance]

- Definiton
- Properties

---
# What are Expectations?

We are familiar with expectations. For example,

- What salary can I expect to earn after graduating from college?
- I will not invest in Facebook because I expect its sock price to fall in the future
- If you complete your homework, you should expect to do well on exams

In statistics, the idea of expectation has a formal, mathematical definition.

Most of econometrics revolves around finding the best .it[conditional expectation] (we'll discuss this concept later in the course)

---
# Expectation: Discrete Case

First we give the mathematical definition of .hi.coral[expectation] for discrete random variables.

Let `$X$` be a discrete random variable with pmf `$p_X(x)$`.

The .hi.purple[expectation] of `$X$`, denoted `$E(X)$` or `$\mu$`, is: 
$$ 
E(X)= \sum_{x \in S} x\cdot p_X(x), 
$$

where `$S$` is the sample space and x is a realization of the random variable `$X$`

The expectation is essentially the .hi[weighted average] of the possible outcomes of `$X$`, or the .it[mean].

---
# Example of Discrete Expectation

Suppose we roll a six-sided .it[unfair] die. The probability of each outcome is provided below:

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      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">P_X(x)</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_right">1</td>
<td class="gt_row gt_right">0.1</td></tr>
    <tr><td class="gt_row gt_right">2</td>
<td class="gt_row gt_right">0.1</td></tr>
    <tr><td class="gt_row gt_right">3</td>
<td class="gt_row gt_right">0.1</td></tr>
    <tr><td class="gt_row gt_right">4</td>
<td class="gt_row gt_right">0.1</td></tr>
    <tr><td class="gt_row gt_right">5</td>
<td class="gt_row gt_right">0.5</td></tr>
    <tr><td class="gt_row gt_right">6</td>
<td class="gt_row gt_right">0.1</td></tr>
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What is the expected outcome of a single throw? By definition:

`\begin{aligned}
  E(X) &= \sum_{x=1}^6 x\cdot p_x(x) \\
  &= (1\cdot 0.1) + (2\cdot 0.1) + (3\cdot 0.1) + (4\cdot 0.1) + (5\cdot 0.5) + (6\cdot 0.1) \\
  &= 4.1
\end{aligned}`

---
# Clicker Question -- Midterm Example
Assume `$X$` can only obtain the values `$1, 2, 3,$` and `$5$`. Given the following PMF, what is `$E(X)$`?

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      <th colspan="2" class="gt_heading gt_title gt_font_normal gt_bottom_border" style><span class='hi slate' style='display: block; margin-bottom: 8px;'>Unfair Die</span></th>
    </tr>
    
  </thead>
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">x</th>
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  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_right">1</td>
<td class="gt_row gt_left">0.1</td></tr>
    <tr><td class="gt_row gt_right">2</td>
<td class="gt_row gt_left">0.2</td></tr>
    <tr><td class="gt_row gt_right">3</td>
<td class="gt_row gt_left">0.2</td></tr>
    <tr><td class="gt_row gt_right">5</td>
<td class="gt_row gt_left">?</td></tr>
  </tbody>
  
  
</table>
</div>

<ol type = "a">
  <li>1.1</li>
  <li>3.6</li>
  <li>3.1</li>
  <li>Cannot be determined from information</li>
</ol>

---
# Expectation: Continuous Case

How would you define an expectation of a continuous random variable?

Let `$Y$` be the continuous random variable with pdf `$f_Y(u)$`. Then the expectation of `$Y$` is

$$
  E(Y)= \int^\infty_{-\infty}y \cdot f_Y(y) dy
$$

This is still a weighted average! `$y$` is the value you observe and `$f_Y(y)$` is the density which is just the relative likelihood or .it[weight].

.it[Don't forget to multiply by y!]

---
# Example of Continuous Expectation

Suppose `$Y$` is a continuous random variable with the pdf `$f_Y(y)=3y^2$` for `$0<y<1$`. Then:

`\begin{equation}
  E(Y) = \int_{-\infty}^{\infty}y\cdot f_Y(y) \ dy = \int_{0}^{1}y \cdot 3y^2 \ dy = \int_{0}^{1} 3y^3 \ dy = \left.\frac{3}{4}y^4 \ \right|_0^1  = \frac{3}{4}
\end{equation}`

---
# Clicker Question

Suppose `$X$` has a .hi.cranberry[uniform distribution] over `$(a,b)$` denoted

`$X \sim U(a,b)$`. That is, `$f_x(x) = \frac{1}{b-a}$` for `$a<x<b$`. What is `$E(X)$`?

.more-left[
<img src="data:image/png;base64,#expectations_files/figure-html/unif-dist-1.svg" width="90%" style="display: block; margin: auto;" />
]

.less-right[
<ol type = "a">
  <li>$\frac{b-a}{2}$</li>
  <li>$\frac{b+a}{2}$</li>
  <li>$\frac{1}{b-a}$</li>
  <li>$1$</li>
</ol>
]

---
# Midterm Example

Consider the probability distribution for random variable X:

$$
  f(x)=.08x,\  0\leq x \leq 5
$$

Find `$E[X]$`

---
# Properties of Expectation
A trick that will make our lives easier is that expectations are a .hi.purple[linear operator] which means you can move constants in and out of the `$E$` function.

Let `$X, Y$` be random variables and `$a, b$` be constants. Then:

- `$E(a) = a$`

- `$E(aX) = aE(X)$`

- `$E(X + b) = E(X) + b$`

- `$E(aX+b) = aE(X) + b$`

- `$E(X + Y) = E(X) + E(Y)$`

---
# Clicker Question
Suppose that the expected income in Boulder is $90,000 and the tax rate is 10%. Then the post-tax income is Y=0.9X where `$X$` is annual income of a Boulder resident. What is the expected post-tax income for a Boulder resident, E(Y)?

---
# Definition of Variance

Recall the variance of a sample is `$$s^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2$$`

We can use expectations to define a similar expression for the population variance.

Let `$X$` be a random variable with `$E(X)< \infty$`. Then the .hi.ruby[variance] of `$X$` is:

$$ 
  Var(X) = E[(X-E(X))^2] 
$$

Imagine writing this out as an integral and solving it. Would not be fun. .it[Instead, we will be using a much simpler equation to calculate variance...]

---
# Alternate Form
Note that we can write,

`\begin{aligned}
Var(X) &= E[ \left(X - E(X) \right)^2 ] = E[ X^2 - 2X\cdot E(X) + [E(X)]^2 ] \\
       &= E(X^2) - 2E(X)\cdot E(X) + [E(X)]^2                                \\
       &= E(X^2) - 2[E(X)]^2 + [E(X)]^2                                      \\
       &= E(X^2) - [E(X)]^2,                                                 
\end{aligned}`

`$$Var(X)=E(X^2)-[E(X)]^2$$` is a much simpler expression of variance (and the one you should use!)

---
# Properties of Expectation
We can compose E with a function of a random variable, g(X).

$$ 
  E[g(X)]= \sum_{x\in S} g(x) \cdot p_x(x)
$$

The result is the expected value of g(X). We will use this property to calculate `$E[X^2]$`

---
# Example

While `$E(X)$` is called the .hi.kelly[first moment], we call `$E(X^2)$` the .hi.coral[second moment] of X.

Therefore, we calculate `$E[X^2]$` using the following equation:

$$ E[X^2] = \sum x^2*p_X(x) $$

or in the continuous case

$$
  \int_{-\infty}^\infty x^2 f_X(x) dx
$$

---
# Example

Let `$X$` be the number of heads in two coin flips.

$$ 
E(X) = 0 \cdot (\frac{1}{4})+ 1 \cdot (\frac{1}{2}) + 2 \cdot (\frac{1}{4}) = \frac{1}{2} + \frac{1}{2} = 1
$$

--
$$ 
E(X^2) = 0^2 \cdot (\frac{1}{4})+ 1^2 \cdot (\frac{1}{2}) + 2^2 \cdot  (\frac{1}{4}) = \frac{1}{2} + 1 = \frac{3}{2} ^\cranberry{*}
$$

This means that 
`$$V(X)=E(X^2)-[E(X)]^2 = \frac{3}{2}-1^2 = \frac{1}{2}$$`

.footnote[.cranberry[<sup>*</sup>].hi[Never] square probabilities when solving $E(X^2)$. Only square $X$ values]

---
# Properties of Variance
There are several important properties of the variance operator:

- `$Var(a) = 0$`

- `$Var(aX) = a^2Var(X)$`

- `$Var(aX+b) = a^2Var(X)$`

- `$Var(aX+bY) = a^2Var(X)+b^2Var(Y)+2ab\cdot Cov(X,Y).$`<sup>.cranberry[*]</sup>

.footnote[.cranberry[<sup>*</sup>] We will discuss more about covariance later, when `$X$` and `$Y$` are independent then the covariance is zero]

---
# Clicker Question
Recall that in Boulder the mean income is $90,000 and the tax rate is 10%. Suppose that the variance of income in Boulder is $1,000. What is the variance of post-tax income, `$Y=0.9X$`?

---
# Midterm Example

Consider the probability distribution for random variable Y

$$
  f(x)=.08x, 0\leq x \leq 5
$$

Find `$V[X]$`:

`\begin{aligned}
  V[X] &= \int_{-\infty}^\infty x^2 * f(x) \ dx - \left[ \int_{-\infty}^\infty x * f(x) \ dx \right]^2 \\
  &= \int_0^5 x^2 * .08x \ dx - \left[\int_0^5 x * .08x \ dx \right]^2= \int_0^5 .08x^3 \ dx - \left[  \int_0^5 .08x^2 \ dx \right]^2 \\
  &= \frac{.08x^4}{4} \ \Big|_0^5 - \left( \frac{.08x^3}{3} \ \Big|_0^5\right) ^2= 12.5-(3.3)^2=1.61 
\end{aligned}`

---
# Midterm Example

Consider the probability distribution for random variable X:

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}

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</style>
<table class="gt_table">
  
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">x</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">P_X(x)</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_right">2</td>
<td class="gt_row gt_left">0.19</td></tr>
    <tr><td class="gt_row gt_right">4</td>
<td class="gt_row gt_left">0.07</td></tr>
    <tr><td class="gt_row gt_right">6</td>
<td class="gt_row gt_left">0.35</td></tr>
    <tr><td class="gt_row gt_right">8</td>
<td class="gt_row gt_left">p</td></tr>
  </tbody>
  
  
</table>
</div>

1. Find the expectation of X
2. Find the variance of X
3. A friend says, the expected value of `$X$` is 5. To justify this he says, "Well `$X$` could be 2,4,6, or 8, so the average is `$\frac{2+4+6+8}{4} = 5$`". Why is your friend wrong? Why isn't `$E[X]=5?$` (1-2) sentences. 
4. Let `$Y\sim N(4,1)$`. Random variable `$W$` is defined as `$W=2X+3Y$`. Given your information about the random variables `$X$` and Y. What is `$E[W]$`? What is `$Var[W]$`? (assuming `$X$` & `$Y$` are independent) }

---
# Midterm Example
Consider the probability distribution for random variable Y:

$$
  f(y)= 8y, 0\leq y \leq \frac{1}{2}
$$

1. Find `$E[Y]$`
2. Find `$P(Y<\frac{1}{3})$`
3. Find `$P(Y=\frac{1}{4})$`
4. Find `$V[Y]$`
5. Find `$P(\frac{3}{4}<Y<1)$`