2.2 — Random Variables & Distributions

ECON 480 • Econometrics • Fall 2022

Dr. Ryan Safner
Associate Professor of Economics

## Contents

Random Variables

Discrete Random Variables

Expected Value and Variance

Continuous Random Variables

The Normal Distribution

# Random Variables

## Experiments

• An experiment is any procedure that can (in principle) be repeated infinitely and has a well-defined set of outcomes

Example

Flip a coin 10 times.

## Random Variables

• A random variable (RV) takes on values that are unknown in advance, but determined by an experiment

• A numerical summary of a random outcome

Example

The number of heads from 10 coin flips

## Random Variables: Notation

• Random variable $X$ takes on individual values $(x_i)$ from a set of possible values

• Often capital letters to denote RV’s

• lowercase letters for individual values

Example

Let $X$ be the number of Heads from 10 coin flips. $\quad x_i \in \{0, 1, 2,...,10\}$

# Discrete Random Variables

## Discrete Random Variables

• A discrete random variable: takes on a finite/countable set of possible values

Example

Let $X$ be the number of times your computer crashes this semester1, $x_i \in \{0, 1, 2, 3, 4\}$

## Discrete Random Variables: Probability Distribution

• Probability distribution of a R.V. fully lists all the possible values of $X$ and their associated probabilities
$x_i$ $P(X=x_i)$
0 0.80
1 0.10
2 0.06
3 0.03
4 0.01

## Discrete Random Variables: pdf

• Probability distribution function (pdf) summarizes the possible outcomes of $X$ and their probabilities

• Notation: $f_X$ is the pdf of $X$:

$f_X=p_i, \quad i=1,2,...,k$

• For any real number $x_i$, $f(x_i)$ is the probablity that $X=x_i$
$x_i$ $P(X=x_i)$
0 0.80
1 0.10
2 0.06
3 0.03
4 0.01
• What is $f(0)$?
• What is $f(3)$?

## Discrete Random Variables: pdf Graph

crashes<-tibble(number = c(0,1,2,3,4),
prob = c(0.80, 0.10, 0.06, 0.03, 0.01))

ggplot(data = crashes) +
aes(x = number,
y = prob)+
geom_col(fill = "#e64173") +
labs(x = "Number of Crashes",
y = "Probability") +
scale_y_continuous(breaks = seq(0,1,0.2),
limits = c(0,1),
expand = c(0,0))+
theme_classic(base_family = "Fira Sans Condensed",
base_size = 20)

## Discrete Random Variables: cdf

• Cumulative distribution function (cdf) lists probability $X$ will be at most (less than or equal to) a given value $x_i$

• Notation: $F_X=P(X \leq x_i)$

$x_i$ $f(x)$ $F(x)$
0 0.80 0.80
1 0.10 0.90
2 0.06 0.96
3 0.03 0.99
4 0.01 1.00
• What is the probability your computer will crash at most once, $F(1)$?
• What about three times, $F(3)$?

## Discrete Random Variables: cdf Graph

crashes <- crashes %>%
mutate(cum_prob = cumsum(prob))

crashes
# A tibble: 5 × 3
number  prob cum_prob
<dbl> <dbl>    <dbl>
1      0  0.8      0.8
2      1  0.1      0.9
3      2  0.06     0.96
4      3  0.03     0.99
5      4  0.01     1   

## Discrete Random Variables: cdf Graph

ggplot(data = crashes) +
aes(x = number,
y = cum_prob) +
geom_col(fill="#e64173") +
labs(x = "Number of Crashes",
y = "Probability") +
scale_y_continuous(breaks = seq(0,1,0.2),
limits = c(0,1),
expand = c(0,0)) +
theme_classic(base_family = "Fira Sans Condensed",
base_size = 20)

# Expected Value and Variance

## Expected Value of a Random Variable

• Expected value of a random variable $X$, written $\mathbb{E}(X)$ (and sometimes $\mu)$, is the long-run average value of $X$ “expected” after many repetitions

$\mathbb{E}(X)=\sum^k_{i=1} p_i x_i$

• $\mathbb{E}(X)=p_1x_1+p_2x_2+ \cdots +p_kx_k$

• A probability-weighted average of $X$, with each $x_i$ weighted by its associated probability $p_i$

• Also called the “mean” or “expectation” of $X$, always denoted either $\mathbb{E}(X)$ or $\mu_X$

## Expected Value: Example I

Example

Suppose you lend your friend $100 at 10% interest. If the loan is repaid, you receive$110. You estimate that your friend is 99% likely to repay, but there is a default risk of 1% where you get nothing. What is the expected value of repayment?

## Expected Value: Example II

Example

Let $X$ be a random variable that is described by the following pdf:

$x_i$ $P(X=x_i)$
1 0.50
2 0.25
3 0.15
4 0.10

Calculate $\mathbb{E}(X)$.

## The Steps to Calculate E(X), Coded

# Make a Random Variable called X
X <- tibble(x_i = c(1,2,3,4), # values of X
p_i = c(0.50,0.25,0.15,0.10)) # probabilities

# Look at tibble
X
# A tibble: 4 × 2
x_i   p_i
<dbl> <dbl>
1     1  0.5
2     2  0.25
3     3  0.15
4     4  0.1 

# Get expected value
X %>%
summarize(expected_value = sum(x_i * p_i))
# A tibble: 1 × 1
expected_value
<dbl>
1           1.85

## Variance of a Random Variable

• The variance of a random variable $X$, denoted $var(X)$ or $\sigma^2_X$ is:

\begin{align*}\sigma^2_X &= \mathbb{E}[(x_i-\mu_X)^2]\\ &=\sum_{i=1}^n(x_i-\mu_X)^2p_i\\ \end{align*}

• This is the expected value of the squared deviations from the mean
• i.e. the probability-weighted average of the squared deviations

## Standard Deviation of a Random Variable

• The standard deviation of a random variable $X$, denoted $sd(X)$ or $\sigma_X$ is:

$\sigma_X=\sqrt{\sigma_X^2}$

• This is the average or expected deviation from the mean

## Standard Deviation: Example I

Example

What is the standard deviation of computer crashes?

$x_i$ $P(X=x_i)$
0 0.80
1 0.10
2 0.06
3 0.03
4 0.01

## The Steps to Calculate sd(X), Coded I

# get the expected value
crashes %>%
summarize(expected_value = sum(number*prob))
# A tibble: 1 × 1
expected_value
<dbl>
1           0.35

# save this for quick use
exp_value <- 0.35

crashes_2 <- crashes %>%
select(-cum_prob) %>% # we don't need the cdf
# create new columns
mutate(deviations = number - exp_value, # deviations from exp_value
deviations_sq = deviations^2, # square deviations
weighted_devs_sq = prob * deviations_sq) # weight squared deviations by probability

## The Steps to Calculate sd(X), Coded II

# look at what we made
crashes_2
# A tibble: 5 × 5
number  prob deviations deviations_sq weighted_devs_sq
<dbl> <dbl>      <dbl>         <dbl>            <dbl>
1      0  0.8       -0.35         0.122           0.098
2      1  0.1        0.65         0.423           0.0423
3      2  0.06       1.65         2.72            0.163
4      3  0.03       2.65         7.02            0.211
5      4  0.01       3.65        13.3             0.133 

## The Steps to Calculate sd(X), Coded III

# now we want to take the expected value of the squared deviations to get variance
crashes_2 %>%
summarize(variance = sum(weighted_devs_sq), # variance
sd = sqrt(variance)) # sd is square root of variance
# A tibble: 1 × 2
variance    sd
<dbl> <dbl>
1    0.648 0.805

## Standard Deviation: Example II

Example

What is the standard deviation of the random variable we saw before?

$x_i$ $P(X=x_i)$
1 0.50
2 0.25
3 0.15
4 0.10

Hint: you already found it’s expected value.

# Continuous Random Variables

## Continuous Random Variables

• Continuous random variables can take on an uncountable (infinite) number of values

• So many values that the probability of any specific value is infinitely small:

$P(X=x_i)\rightarrow 0$

• Instead, we focus on a range of values it might take on

## Continuous Random Variables: pdf I

• Probability density function (pdf) of a continuous variable represents the probability between two values as the area under a curve

• The total area under the curve is 1

• Since $P(a)=0$ and $P(b)=0$, $P(a<X<b)=P(a \leq X \leq b)$

• See today’s appendix for how to graph math/stats functions in ggplot!

Example

$P(0 \leq X \leq 2)$

## Continuous Random Variables: pdf II

• FYI using calculus:

$P(a \leq X \leq b) = \int_a^b f(x) dx$

• Complicated: software or (old fashioned!) probability tables to calculate

Example

$P(0 \leq X \leq 2)$

## Continuous Random Variables: cdf I

• The cumulative density function (cdf) describes the area under the pdf for all values less than or equal to (i.e. to the left of) a given value, $k$

$P(X \leq k)$

Example

$P(X \leq 2)$

## Continuous Random Variables: cdf II

• Note: to find probability of values greater than or equal to (to the right of) a given value $k$:

$P(X \geq k)=1-P(X \leq k)$

Example

$P(X \geq 2) = 1 - P(X \leq 2)$

$P(X \geq 2)=$ area under the pdf curve to the right of 2

# The Normal Distribution

## The Normal Distribution

• The Gaussian or normal distribution is the most useful type of probability distribution

$X \sim N(\mu,\sigma)$

• $X$ is distributed Normally with mean $\mu$ and standard deviation $\sigma$

• Continuous, symmetric, unimodal

## The Normal Distribution: pdf

• FYI: The pdf of $X \sim N(\mu, \sigma)$ is

$P(X=k)= \frac{1}{\sqrt{2\pi \sigma^2}}e^{-\frac{1}{2}\big(\frac{(k-\mu)}{\sigma}\big)^2}$

• Do not try and learn this, we have software and (previously tables) to calculate pdfs and cdfs

## The Standard Normal Distribution

• The standard normal distribution (often referred to as $Z$) has mean 0 and standard deviation 1

$Z \sim N(0,1)$

## The Standard Normal cdf

• The standard normal cdf, often referred to as $\Phi$:

$\Phi(k)=P(Z \leq k)$

(again, the area under the pdf curve to the left of some value $k$)

## The 68-95-99.7 Empirical Rule

• 68-95-99.7% empirical rule: for a normal distribution:

## The 68-95-99.7 Empirical Rule

• 68-95-99.7% empirical rule: for a normal distribution:

• $P(\mu-1\sigma \leq X \leq \mu+1\sigma) \approx$ 68%

## The 68-95-99.7 Empirical Rule

• 68-95-99.7% empirical rule: for a normal distribution:

• $P(\mu-1\sigma \leq X \leq \mu+1\sigma) \approx$ 68%

• $P(\mu-2\sigma \leq X \leq \mu+2\sigma) \approx$ 95%

## The 68-95-99.7 Empirical Rule

• 68-95-99.7% empirical rule: for a normal distribution:

• $P(\mu-1\sigma \leq X \leq \mu+1\sigma) \approx$ 68%

• $P(\mu-2\sigma \leq X \leq \mu+2\sigma) \approx$ 95%

• $P(\mu-3\sigma \leq X \leq \mu+3\sigma) \approx$ 99.7%

• 68/95/99.7% of observations fall within 1/2/3 standard deviations of the mean

## Standardizing Normal Distributions

• We can take any normal distribution (for any $\mu, \sigma)$ and standardize it to the standard normal distribution by taking the Z-score of any value, $x_i$:

$Z=\frac{x_i-\mu}{\sigma}$

• Subtract any value by the distribution’s mean and divide by standard deviation

• $Z$: number of standard deviations $x_i$ value is away from the mean

## Standardizing Normal Distributions: Example I

Example

On August 8, 2011, the Dow dropped 634.8 points, sending shock waves through the financial community. Assume that during mid-2011 to mid-2012 the daily change for the Dow is normally distributed, with the mean daily change of 1.87 points and a standard deviation of 155.28 points. What is the $Z$-score?

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{634.8-1.87}{155.28}$

$Z = -4.1$

This is 4.1 standard deviations $(\sigma)$ beneath the mean, an extremely low probability event.

## Standardizing Normal Distributions: Example II

Example

In the last quarter of 2021, a group of 64 mutual funds had a mean return of 2.4% with a standard deviation of 5.6%. These returns can be approximated by a normal distribution.

What percent of the funds would you expect to be earning between -3.2% and 8.0% returns?

Convert to standard normal to find $Z$-scores for $8$ and $-3.2.$

$P(-3.2 < X < 8)$

$P(\frac{-3.2-2.4}{5.6} < \frac{X-2.4}{5.6} < \frac{8-2.4}{5.6})$

$P(-1 < Z < 1)$

$P(X \pm 1\sigma)=0.68$

## Standardizing Normal Distributions: Example III

Example

In the last quarter of 2021, a group of 64 mutual funds had a mean return of 2.4% with a standard deviation of 5.6%. These returns can be approximated by a normal distribution.

1. What percent of the funds would you expect to be earning 2.4% or less?

2. What percent of the funds would you expect to be earning between -8.8% and 13.6%?

3. What percent of the funds would you expect to be earning returns greater than 13.6%?

## Finding Z-score Probabilities I

Probability to the left of $z_i$

$P(Z \leq z_i)= \underbrace{\Phi(z_i)}_{\text{cdf of }z_i}$

Probability to the right of $z_i$

$P(Z \geq z_i)= 1-\underbrace{\Phi(z_i)}_{\text{cdf of }z_i}$

## Finding Z-score Probabilities II

Probability between $z_1$ and $z_2$

$P(z_1 \geq Z \geq z_2)= \underbrace{\Phi(z_2)}_{\text{cdf of }z_2} - \underbrace{\Phi(z_1)}_{\text{cdf of }z_1}$

## Finding Z-score Probabilities III

• pnorm() calculates probabilities with a normal distribution with arguments:
• x = the value
• mean = the mean
• sd = the standard deviation
• lower.tail =
• TRUE if looking at area to LEFT of value
• FALSE if looking at area to RIGHT of value

## Finding Z-score Probabilities IV

Example

Let the distribution of grades be normal, with mean 75 and standard deviation 10.

• Probability a student gets at least an 80
pnorm(80,
mean = 75,
sd = 10,
lower.tail = FALSE) # looking to right
[1] 0.3085375

## Finding Z-score Probabilities V

Example

Let the distribution of grades be normal, with mean 75 and standard deviation 10.

• Probability a student gets at most an 80
pnorm(80,
mean = 75,
sd = 10,
lower.tail = TRUE) # looking to left
[1] 0.6914625

## Finding Z-score Probabilities VI

Example

Let the distribution of grades be normal, with mean 75 and standard deviation 10.

• Probability a student gets between 65 and 85
# subtract two left tails!
pnorm(85, # larger number first!
mean = 75,
sd = 10,
lower.tail = TRUE) - # looking to left, & SUBTRACT
pnorm(65, # smaller number second!
mean = 75,
sd = 10,
lower.tail = TRUE) #looking to left
[1] 0.6826895