Introduction

R includes a number of operations that associate distributional families (i.e., normal, binomial) with a set of four functions. These functions are:

These letter prefixes combine with the names of distribution, with additional arguments. For example, the generate random normal numbers, we take the prefix r with the distribution norm to get the function rnorm. He, we demonstrate generating 5 random normal numbers

## Generate 5 random numbers with N(0, 1)
rnorm(n = 5, mean = 0, sd = 1)
## [1] -0.090412  0.263064  0.360291 -0.390299 -2.671451

Note that this function takes as arguments the mean and standard deviation, the two distributional parameters associated with a normal distribution.

You can find the full list of available distributions with ?Distributions

Question 1: Using the documentation effectively is an important component of improving your efficiency and effectiveness with R. Copy and paste ?rbinom into your console and read the documentation. What does this function do? Explain the difference between rbinom(5, 1) and rbinom(1, 5). Which of these would you use to simulate a collection of Bernoulli random variables?

Density function

The density function (d) is the R implementation of the PMF/PDF functions for distributions. Given a realized value, along with distributional parameters, these functions will return either a probability (for discrete distributions) or a density (for continuous). For example, suppose we have \(Y \sim Bin(5, \pi = .25)\). We can find the probability that in 5 trials we have 3 success is

\[ P(Y = 3) = \binom{5}{3} (0.25)^3(0.75)^{5-3} \] We can find this directly with

dbinom(x = 3, size = 5, prob = 0.25)
## [1] 0.087891

Cumulative function

Similarly, we have the cumulative density/mass functions (p) that tells use the probability of observing values greater than or less than our argument (i.e., this function can be used to compute p-values)

Considering our previous situation with \(Y ~ Bin(5, \pi = .25)\), consider

\[ P(Y \leq 3) = \sum_{i=0}^3 \binom{5}{i} (0.25)^i(0.75)^{5-i} \] We can find this probability using pbinom or, slightly more laboriously, using dbinom

pbinom(3, 5, 0.25)
## [1] 0.98438
## These are the same
# Note the vector argument x = 0:3
sum(dbinom(x = 0:3, size = 5, prob = 0.25))
## [1] 0.98437

It follows from this that we can find \(P(Y > 3)\) with

1 - pbinom(3, 5, 0.25)
## [1] 0.015625

Question 2: Use pbinom function to determine \(P(2 \leq Y \leq 4)\) when \(Y ~ Bin(5, \pi = .25)\)

Question 3: Consider a random variable \(Y \sim Pois(\mu = 2)\). How would you use a probability function to find \(P(Y > 3)\)?

Quantile function

Question 4: What is the q prefix used for? Use the appropriate function to find the 95% CI critical values for a t distribution with 20 degrees of freedom