2.2 — Random Variables & Distributions — Appendix

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 in R

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 and Statistical Functions in R

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)
# x values are integers 1 through 10
ggplot(data = tibble(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 = tibble(x = 1:10))+
  aes(x = x)+
  stat_function(fun = my_function)

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

ggplot(data = tibble(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:

Action/Type Prefix
random draw r
density (pdf) d
cumulative density (cdf) p
quantile (inverse cdf) q

Distributions:

Distribution Name in R
Normal norm
Uniform unif
Student’s t t
Binomial binom
Negative binomial nbinom
Hypergeometric hyper
Weibull weibull
Beta beta
Gamma gamma

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

Some common examples:

  1. Take random draws from a normal distribution:
rnorm(n = 10, # take 10 draws from a normal distribution with:
      mean = 2, # mean of 2
      sd = 1) # sd of 1
 [1] 3.4582468 2.0740456 0.9092114 1.1790483 2.1474758 3.0726526 3.6894126
 [8] 3.0430861 1.9429537 0.5569242
  1. Get probability of a random variable being less than or equal to a value (cdf) from a normal distribution:
# 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
  1. Find the value of a distribution that is a specified percentile.
# 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 = tibble(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 = tibble(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 = tibble(x = -4:4))+
  aes(x = x)+
  # graph the curve
  stat_function(fun = dnorm,
                color = "blue",
                size = 2,
                linetype = "dashed")+
  # shade area under curve (between -2 and 2)
  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),
                geom = "area",
                size = 2,
                fill = "gray",
                alpha = 0.5)+
  stat_function(fun = dnorm,
                args = list(mean = 75, sd = 10),
                geom = "area",
                xlim = c(65,85),
                fill = "#e64173")+
  labs(x = "X",
       y = "Probability")+
  scale_x_continuous(breaks = seq(35,115,5))+
    scale_y_continuous(limits = c(0,0.045),
                     expand = c(0,0))+
  theme_classic(base_family = "Fira Sans Condensed",
           base_size = 20)