Introduction to Binomial Distribution Functions

• dbinom(): Probability mass function
• pbinom(): Cumulative distribution function
• qbinom(): Quantile function
• rbinom(): Random number generation

These functions are essential for working with the binomial distribution in R.

dbinom(): Probability Mass Function

• Purpose: Computes the probability of getting exactly k successes in n trials.
• Syntax: dbinom(k, size = n, prob = p)

k <- 0:10
n <- 10
p <- 0.5
probabilities <- dbinom(k, size = n, prob = p)
barplot(probabilities, names.arg = k, main = "dbinom(): Binomial Probability Mass Function", xlab = "Number of Successes (k)", ylab = "Probability", col = "lightblue")

pbinom(): Cumulative Distribution Function

• Purpose: Computes the cumulative probability of getting up to k successes in n trials.
• Syntax: pbinom(q, size = n, prob = p)

k <- 0:10
n <- 10
p <- 0.5
cumulative_probabilities <- pbinom(k, size = n, prob = p)
plot(k, cumulative_probabilities, type = "s", main = "pbinom(): Binomial Cumulative Distribution Function", xlab = "Number of Successes (k)", ylab = "Cumulative Probability", col = "blue")

qbinom(): Quantile Function

• Purpose: Computes the smallest number of successes k such that the cumulative probability is at least p.
• Syntax: qbinom(p, size = n, prob = p)

probabilities <- seq(0, 1, length.out = 100)
n <- 10
p <- 0.5
quantiles <- qbinom(probabilities, size = n, prob = p)
plot(probabilities, quantiles, type = "s", main = "qbinom(): Binomial Quantile Function", xlab = "Probability", ylab = "Quantile (k)", col = "darkgreen")

rbinom(): Random Number Generation

• Purpose: Generates random numbers from a binomial distribution.
• Syntax: rbinom(n, size = n, prob = p)

set.seed(123)
random_numbers <- rbinom(1000, size = 10, prob = 0.5)
hist(random_numbers, breaks = seq(-0.5, 10.5, by = 1), main = "rbinom(): Random Numbers from Binomial Distribution", xlab = "Number of Successes (k)", col = "lightgreen")

Summary

• dbinom(): Gives the probability of exactly k successes.
• pbinom(): Gives the cumulative probability of up to k successes.
• qbinom(): Finds the smallest k for a given cumulative probability.
• rbinom(): Generates random numbers from a binomial distribution.

These functions are fundamental for working with binomial experiments in R.