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The binomial distribution is a fundamental discrete probability distribution. It describes the number of successes in a fixed number of Bernoulli trials, with the same probability of success.
Let's walk through the basics of the binomial distribution in R:
1. dbinom, pbinom, qbinom, and rbinom
R has a set of functions for the binomial distribution:
dbinom(x, size, prob): Gives the probability of getting x successes.pbinom(q, size, prob): Cumulative distribution function. Gives the probability of getting q or fewer successes.qbinom(p, size, prob): Quantile function. Gives the number of successes for a given cumulative probability p.rbinom(n, size, prob): Generates n random values based on the binomial distribution.2. Examples
Let's say we're flipping a coin (fair coin, so p = 0.5) 10 times, and we want to know the probability of getting exactly 5 heads:
dbinom(5, size=10, prob=0.5)
What is the probability of getting 5 or fewer heads?
pbinom(5, size=10, prob=0.5)
What's the number of heads you'd expect to get 90% of the time (i.e., the 90th percentile)?
qbinom(0.9, size=10, prob=0.5)
Generate 5 random values where we flip a coin 10 times:
rbinom(5, size=10, prob=0.5)
3. Plotting the Distribution
Plotting the probability for each number of successes from 0 to 10:
x <- 0:10 y <- dbinom(x, size=10, prob=0.5) plot(x, y, type="h", main="Binomial Distribution", xlab="Number of successes", ylab="Probability", col="blue", lwd=2) points(x, y, pch=16, col="red")
The above code plots the probability mass function for a binomial distribution with 10 trials and a 0.5 probability of success.
Remember, the binomial distribution assumes that each trial is independent, and the probability of success remains constant across trials. If these assumptions are not met, then the binomial distribution might not be appropriate.
R Binomial Distribution Example:
# Generate a sample from a binomial distribution set.seed(123) sample_data <- rbinom(100, size = 10, prob = 0.5)
How to Generate Binomial Random Variables in R:
# Generate binomial random variables set.seed(123) binomial_var <- rbinom(5, size = 10, prob = 0.3)
Probability Mass Function of Binomial Distribution in R:
# Probability mass function (PMF) of binomial distribution x <- 0:10 pmf_values <- dbinom(x, size = 10, prob = 0.5)
Binomial Distribution Functions in R:
# Cumulative distribution function (CDF) of binomial distribution x <- 0:10 cdf_values <- pbinom(x, size = 10, prob = 0.5)
Plotting Binomial Distribution in R:
# Plot binomial distribution
x <- 0:10
pmf_values <- dbinom(x, size = 10, prob = 0.5)
plot(x, pmf_values, type = "h", lwd = 2, col = "blue", ylim = c(0, 0.3),
main = "Binomial Distribution", xlab = "Number of Successes", ylab = "Probability")
Fitting Binomial Distribution to Data in R:
# Fit binomial distribution to data data <- c(3, 4, 5, 6, 7) fit <- fitdistr(data, "binomial", list(size = 10, prob = 0.5))
Calculating Cumulative Probabilities in Binomial Distribution in R:
# Calculate cumulative probabilities x <- 0:10 cumulative_prob <- pbinom(x, size = 10, prob = 0.5)
Confidence Intervals for Binomial Proportions in R:
# Calculate confidence interval for binomial proportion binom_conf_interval <- binom.test(x = 20, n = 30, conf.level = 0.95)
Hypothesis Testing with Binomial Distribution in R:
# Perform hypothesis testing with binomial distribution binom_test_result <- binom.test(x = 15, n = 20, p = 0.5, alternative = "two.sided")
Comparing Binomial Distributions in R:
# Compare two binomial distributions group1 <- rbinom(50, size = 10, prob = 0.4) group2 <- rbinom(50, size = 10, prob = 0.6) t_test_result <- t.test(group1, group2)