• A discrete variable is a random variable with finite values in a limited range
  
• usually represented through integers
  
• The probability of a discrete variable is called a probability mass function(pmf)

Example:
- The random variable X is the number of true or false questions you get correct on the test
- this is a discrete random variable since the range of X is finite and there is a finite interval of its range

• A continuous variable is a random variable which takes values from an infinite and uncountable set
  
• usually represented by rounded real numbers
  
• the probability of a continuous variable is called a probability density function(pdf)

Examples of continuous variables:
- height
- weight
- time

• Discrete Uniform
  PMF:\(P(X=k) = \frac{1}{n} \qquad \text{for } k=1,2,\dots,n\)

• Binomial
  PMF:\(P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \qquad \text{for } k=0,1,\dots,n\)

• Poisson
  PMF:\(P(X=k) = e^{-\lambda} \frac{\lambda^k}{k!} \qquad \text{for } k=0,1,2,\dots\)

a <- 1
b <- 10

x <- seq(a, b, by = 1)

y <- dunif(x, min = a, max = b)

plot_ly(x = x, y = y, type = "bar", name = "Discrete Uniform Distribution") %>%
  layout(title = "Discrete Uniform Distribution",
         xaxis = list(title = "X"),
         yaxis = list(title = "Probability Mass"))

• Continuous Uniform
  \(f(x) = \begin{cases}\frac{1}{b-a} & \text{for } a \leq x \leq b \\0 & \text{otherwise}\end{cases}\)
  
• Normal
  \(f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}\)
  
• Standard Normal
  \(f(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}\)
  
• Exponential
  \(f(x) = \begin{cases}\lambda e^{-\lambda x} & \text{for } x \geq 0 \\0 & \text{for } x < 0\end{cases}\)

x <- seq(-4, 4, length.out = 1000)

y <- dnorm(x)

plot_ly(x = x, y = y, type = "scatter", mode = "lines",
        name = "Standard Normal Density") %>%
  layout(title = "Standard Normal Distribution",
         xaxis = list(title = "Z-Score"),
         yaxis = list(title = "Density"))

library(ggplot2)
data <- data.frame(
  animal = c("Mammals", "Fish", "Birds", "Reptiles", "Amphibians"),
  count = c(20, 25, 5, 30, 10)
)

ggplot(data, aes(x = animal, y = count, fill = animal)) +
  geom_bar(stat = "Type of Animals") +
  xlab("Animals") +
  ylab("Count")

data <- data.frame(
  height = rnorm(1000, mean = 80, sd = 5)
)

ggplot(data, aes(x = height)) +
  geom_histogram(binwidth = 0.5, fill = "red", color = "white") +
  xlab("Height (in)") +
  ylab("Count")

• Both graphs show how a discrete variable and a continuous varibale distribution looks like. For the discrete, it shows categories on the x axis and the count on the y, whereas the the continuous variable shows the infinite range of height which forms and standard normal curve.

• Overall this information shows how discrete and continuous variables are different. It also gives an idea on which types of data apply to discrete variables and which types of data apply to continuous variables. Additionally the graphs and distributions show how the random variables look like when using a specific distribution.