1. Statistical Distributions in R

• R has many built-in statistical distributions
  • e.g., binomial, poisson, normal, chi square, …
• Each distribution in R has four functions:
  • These functions begin with a "d", "p", "q", or "r" and are followed by the name of the distribution
• ddist(): gives the density of the distribution
• rdist(): generates random numbers out of the distribution
• qdist(): gives the quantile of the distribution
• pdist(): gives the cumulative distribution function (CDF)

• Consider tossing a coin 10 times
• The probability distribution for the two possible outcomes follows a binomial distribution
• Let's calculate the probability of getting five heads using the function dbinom()

str(dbinom) # binomial probability mass func

## function (x, size, prob, log = FALSE)

dbinom(5, 10, 0.5) # Pr[X = 5] = ?

## [1] 0.2460938

• Next, let's calculate the probability of getting 5 or fewer heads using the function pbinom()

str(pbinom) # binomial CDF 

## function (q, size, prob, lower.tail = TRUE, log.p = FALSE)

pbinom(5, 10, 0.5) # Pr[X <= 5] = ?

## [1] 0.6230469

• Now, suppose we have the probability 0.75 and we want to calculate the number of heads whose CDF is equal to that using qnorm() (note that this is the inverse of pnorm())

str(qbinom) # binomial quantile func

## function (p, size, prob, lower.tail = TRUE, log.p = FALSE)

qbinom(0.75, 10, 0.5) # get the value of ? s.t. Pr[X <= ?] = 0.75

## [1] 6

• Finally, let's generate 20 independent samples from a binomial(10, 0.5). This is equivalent to repeatedly (i.e., 20 times) flipping a coin 10 times and counting the number of heads.

str(rbinom) # binomial random number generator

## function (n, size, prob)

rbinom(20, 10, 0.5) # 20 ind samples from binomial(10, 0.5)

##  [1] 4 4 5 4 6 8 7 5 6 3 4 4 6 6 4 4 6 4 5 7

• Calculate the value of the probability density function at \(X = 0\)

str(dnorm) # normal pdf

## function (x, mean = 0, sd = 1, log = FALSE)

dnorm(x = 0, mean = 0, sd = 1)

## [1] 0.3989423

• Calculate the probability that \(X \leq 0\)

str(pnorm) # normal CDF

## function (q, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)

pnorm(0, mean = 0, sd = 1) # Pr[X <= 0] = ?

## [1] 0.5

• Find the value for which the CDF = 0.975

str(qnorm) # normal quantile func

## function (p, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)

qnorm(0.975, mean = 0, sd = 1) # PR[X <= ?] = 0.975

## [1] 1.959964

• Generate 10 independent random numbers from a standard normal distribution

str(rnorm) # generate random number from normal dist

## function (n, mean = 0, sd = 1)

rnorm(10, mean = 0, sd = 1)

##  [1]  0.28732831  1.28870120  0.95492333 -1.05616106  0.71571889
##  [6] -0.06861534  0.25125574  1.13770718  0.31197057  1.73683798

Let's try plotting a normal curve (more on plotting later)

x <- seq(from = -3, to = 3, by = 0.05)
y <- dnorm(x, mean = 0, sd = 1)
plot(x, y, type = "l")