2.2 — Random Variables and Distributions — Class Content

Problem Set 1

Problem Set 1 is posted and is due by class Tuesday September 14. Please see the instructions for more information on how to submit your assignment (there are multiple ways!).

Problem set 2 (on classes 2.1-2.2) will be posted shortly, and will be due by Tuesday September 21.

Properties of Expected Value and Variance

There are several useful mathematical properties of expected value and variance.

Property 1: the expected value of a constant is itself, and the variance of a constant is 0.

\[\begin{align*} E(c)&=c\\ var(c)&=0\\ sd(c)&=0\\ \end{align*}\]

For any constant, \(c\)

• Example: \(E(2)=2\), \(var(2)=0\), \(sd(2)=0\)

Property 2: adding or subtracting a constant to a random variable and then taking the mean or variance:

\[\begin{align*} E(X \pm c)&=E(X) \pm c\\ var(X \pm c)&=X\\ sd(X \pm c)&=X\\ \end{align*}\]

For any constant, \(c\)

• Example: \(E(2+X)=2+E(X)\), \(var(2+X)=var(X)\), \(sd(2+X)=sd(X)\)

Property 3: multiplying a constant to a random variable and then taking the mean or variance:

\[\begin{align*} E(aX)&=E(X) aE(X)\\ var(aX)&=a^2var(X)\\ sd(aX)&=|a|sd(X)\\ \end{align*}\]

For any constant, \(a\)

• Example: \(E(2X)=2E(X)\), \(var(2X)=4var(X)\), \(sd(2X)=2sd(X)\)

Property 4: the expected value of the sum of two random variables is equal to the sum of each random variable’s expected value:

\[E(X \pm Y)=E(X) \pm E(Y)\]

Creating Mathematical Functions

You can create custom mathematical functions using mosaic by defining an R function() with multiple arguments. As a simple example, make the function \(f(x) = 10x-x^2\) (with one argument, \(x\) since it is a univariate function) as follows:

# store as a named function, I'll call it "my_function"
my_function<-function(x){10*x-x^2}

# look at it
my_function

## function(x){10*x-x^2}

There are some notational requirements from R for making functions. Any coefficient in front of a variable (such as the 10 in 10x must be explicitly multiplied by the variable, as in 10*x).

To use the function to calculate its value at a particular value of x, simply define what the (x) is and run your named function on it:

# f of 2 
my_function(2)

## [1] 16

# f of 2 and 4
my_function(c(2,4))

## [1] 16 24

# f of 2 through 7
my_function(2:7)

## [1] 16 21 24 25 24 21

# ALTERNATIVELY
# define x first as a vector and then run function on it

x<-c(2,4)
my_function(x)

## [1] 16 24

Graphing Mathematical Functions

In ggplot there is a dedicated stat_function() (equivalent to a geom_ layer) to graph mathematical and statistical functions. All that is needed is a data.frame of a range of x values to act as the source for data, and set x equal to those values for aesthetics.

library(tidyverse)

## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.1 ──

## ✓ ggplot2 3.3.5     ✓ purrr   0.3.4
## ✓ tibble  3.1.4     ✓ dplyr   1.0.7
## ✓ tidyr   1.1.3     ✓ stringr 1.4.0
## ✓ readr   2.0.0     ✓ forcats 0.5.1

## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## x dplyr::filter() masks stats::filter()
## x dplyr::lag()    masks stats::lag()

# x values are integers 1 through 10
ggplot(data = data.frame(x = 1:10))+
  aes(x = x)

Then we add the stat_function, where fun = is the most important argument where you define the to function to graph as your function created above, for example, our my_function.

ggplot(data = data.frame(x = 1:10))+
  aes(x = x)+
  stat_function(fun = my_function) 

You can also adjust things like size, color, and line type.

ggplot(data = data.frame(x = 1:10))+
  aes(x = x)+
  stat_function(fun = my_function, color = "blue", size = 2, linetype = "dashed") 

Bult-in Statistical Functions

There are some standard statistical distributions built into R. They require a combination of a specific prefix and a distribution.

Prefixes:

Distributions:

Thus, what you want is a combination of the prefix and the distribution.

rnorm(n = 10, # take 10 draws from a normal distribution with:
      mean = 2, # mean of 2
      sd = 1) # sd of 1

##  [1] 0.9306446 1.9587986 1.7016603 1.2481484 1.3399839 1.9677189 0.3313595
##  [8] 1.6183993 2.5601664 0.5277235

# find probability of area to the LEFT of a number on pdf (note this = cdf of that number!)
pnorm(q = 80, # number is 80 from a distribution where: 
      mean = 200, # mean is 100
      sd = 100, # sd is 100
      lower.tail = TRUE) # looking to the LEFT in lower tail

## [1] 0.1150697

# find the 38th percentile value
qnorm(p = 0.38, # 38th percentile from a distribution where:
      mean = 200, # mean is 200
      sd = 100) # sd is 100

## [1] 169.4519

Graphing Statistical Functions

You can also graph these commonly used statistical functions by setting fun = the named functions in your stat_function() layer. If you want to specify the mean and standard deviation, use args = list() to include the required arguments from the named function above (e.g. dnorm needs mean and sd).

ggplot(data = data.frame(x = -400:600))+
  aes(x = x)+
  stat_function(fun = dnorm, args = list(mean = 200, sd = 200), color = "blue", size = 2, linetype = "dashed") 

If you don’t include this, it will graph the standard distribution:

ggplot(data = data.frame(x = -4:4))+
  aes(x = x)+
  stat_function(fun = dnorm, color = "blue", size = 2, linetype = "dashed") 

To add shading under a distribution, simply add a duplicate of the stat_function() layer, but add geom="area" to indicate the area beneath the function should be filled, and you can limit the domain of the fill with xlim=c(start,end), where start and end are the x-values for the endpoints of the fill.

# graph normal distribution and shade area between -2 and 2
ggplot(data = data.frame(x = -4:4))+
  aes(x = x)+
  stat_function(fun = dnorm, color = "blue", size = 2, linetype = "dashed")+
  stat_function(fun = dnorm, xlim = c(-2,2), geom = "area", fill = "green", alpha=0.5)

Hence, here is one graph from my slides:

ggplot(data = tibble(x=35:115))+
  aes(x = x)+
  stat_function(fun = dnorm, args = list(mean = 75, sd = 10), size = 2, color="blue")+
  stat_function(fun = dnorm, args = list(mean = 75, sd = 10), geom = "area", xlim = c(65,85), fill="blue", alpha=0.5)+
  labs(x = "X",
       y = "Probability")+
  scale_x_continuous(breaks = seq(35,115,5))+
  theme_classic(base_family = "Fira Sans Condensed",
           base_size=20)