# 1. Introduction

R has a number of built in functions for calculations involving probability distributions, both discrete and continuous, including the binomial, normal, Poisson, geometric, gamma, beta, and others we have seen.

For each distribution, R has four primary functions. Each function has a one letter prefix followed by the root name of the function. The names make mnemonic sense for continuous random variables but are used in both cases. For example dnorm is the height of the density of a normal curve while dbinom returns the probability of an outcome of a binomial distribution. Here is a table of these commands.

Prefix Condinuous Discrete
d density probability (pmf)
p probability (cdf) probability (cdf)
q quantile quantile
r random random

Distribution root
Binomial binom
Geometric geom
Hypergeometric hyper
Poisson pois
Normal norm
Gamma gamma
Beta beta
t t
F f
Chi-squre chisq
Tukey’s HSD tukey

# 2. normal distribution

Function Purpose Example
dnorm(x, mean, sd) Probability Density Function (PDF) dnorm(0.1, 0, 1) gives the density (height of the PDF) of the normal 0.1 with mean = 0 and sd = 1
pnorm(q, mean, sd) Cumulative Distribution Function (CDF) pnorm(1.96, 0, 1) gives the area under the standard normal curve to the left of 1.96, i.e., 0.975.
qnorm(q, mean, sd) Quantile Function - inverse of pnorm qnorm(0.975, 0, 1) gives the value at which the CDF of the standard normal is 0.975, i.e., 1.96
rnorm(n, mean, sd) Generate random numbers from normal distribution rnorm(10, 0, 1) generate 10 numbers from a normal with mean = 0 and sd =1

Here are four examples,

dnorm(0.1, 0, 1)
## [1] 0.3969525
pnorm(1.96, 0, 1)
## [1] 0.9750021
qnorm(0.975, 0, 1)
## [1] 1.959964
rnorm(10, 0, 1)
##  [1]  0.71147273  1.41254191 -0.01079530  0.34716732  0.52182994
##  [6]  0.77965795 -1.36593175  0.25465049 -0.20048947  0.04732296

# 3. binomial distribution

Function Purpose Example
dbinom(x, size, prob) Probability Density Function (PDF) dbinom(1, 2, 0.5) gives the density (height of the PDF) of the binomial 1 with size = 2 and prob = 0.5
pbinom(q, size, prob) Cumulative Distribution Function (CDF) pbinom(1, 2, 0.5) gives the area under the binomial curve to the left of 1, i.e., 0.75.
qbinom(p, size, prob) Quantile Function - inverse of pnorm qbinom(0.75, 2, 0.5) gives the value at which the CDF of the binomial is 0.75, i.e., 1.5
rbinom(n, size, prob) Generate random numbers from normal distribution rbinom(20, 2, 0.5) generate 4 numbers from a binomial with size = 2 and probability = 0.5

Here are four examples,

dbinom(1, 2, 0.5)
## [1] 0.5
pbinom(1, 2, 0.5)
## [1] 0.75
qbinom(0.75, 2, 0.5)
## [1] 1
rbinom(20, 2, 0.5)
##  [1] 0 2 2 1 0 1 1 2 2 1 0 1 1 1 1 0 1 0 0 1