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Compute the Negative Binomial Density in R

The negative binomial distribution is a discrete probability distribution that describes the number of successes in a sequence of independent Bernoulli trials before a specified number of failures occur. In R, the dnbinom() function calculates the negative binomial probability density (or mass function, since it's a discrete distribution).

Here's a tutorial on how to compute the negative binomial density in R:

1. Basic Syntax

The basic syntax for dnbinom() is:

dnbinom(x, size, prob)

Where:

  • x is a vector of quantiles.
  • size is the target number of successes.
  • prob is the probability of success on each trial.

2. Examples

a. Basic Example:

Let's find the probability of getting 5 successes before 3 failures occur, given the probability of success on any given trial is 0.7:

x <- 5
size <- 3
prob <- 0.7

dnbinom(x, size, prob)

b. Vector of Quantiles:

To find the probabilities for a range of successes, you can provide a vector for x:

x <- 0:10
probabilities <- dnbinom(x, size, prob)
print(probabilities)

3. Plotting the Negative Binomial Density

Visualizing the distribution can be helpful:

library(ggplot2)

df <- data.frame(x = 0:10, y = dnbinom(0:10, size, prob))

ggplot(df, aes(x, y)) + 
  geom_bar(stat="identity") + 
  labs(title = "Negative Binomial Distribution",
       x = "Number of Successes", 
       y = "Probability") + 
  theme_minimal()

4. Mean and Variance

For the negative binomial distribution:

  • Mean: μ=probsize​
  • Variance: σ2=prob2size×(1−prob)​

You can compute these in R:

mean_nb <- size / prob
variance_nb <- (size * (1 - prob)) / (prob^2)

print(mean_nb)
print(variance_nb)

Summary:

The dnbinom() function allows for easy computation of negative binomial probabilities in R. By understanding the underlying concepts and how to use the function, you can efficiently analyze and visualize negative binomial data.

  1. R Compute Negative Binomial Density Example:

    # Example of computing negative binomial density
x <- 5
size <- 2
prob <- 0.3
neg_binom_density <- dnbinom(x, size = size, prob = prob)

  2. How to Calculate Negative Binomial Probability in R:

    Use the dnbinom function to calculate the probability.

    # Calculate negative binomial probability
x <- 5
size <- 2
prob <- 0.3
neg_binom_prob <- dnbinom(x, size = size, prob = prob)

  3. dnbinom Function in R for Negative Binomial Density:

    Use the dnbinom function for negative binomial density.

    # Using dnbinom for negative binomial density
x <- 5
size <- 2
prob <- 0.3
neg_binom_density <- dnbinom(x, size = size, prob = prob)

  4. Negative Binomial Distribution in R:

    Create a negative binomial distribution using the rnbinom function.

    # Negative binomial distribution
n_samples <- 1000
size <- 2
prob <- 0.3
neg_binom_dist <- rnbinom(n_samples, size = size, prob = prob)

  5. Probability Mass Function for Negative Binomial Distribution in R:

    Plot the probability mass function for the negative binomial distribution.

    # Probability mass function for negative binomial distribution
x_values <- 0:20
pmf_values <- dnbinom(x_values, size = size, prob = prob)
plot(x_values, pmf_values, type = "h", lwd = 2, col = "blue", main = "Negative Binomial PMF")

  6. Generating Random Numbers from Negative Binomial Distribution in R:

    Generate random numbers from the negative binomial distribution.

    # Generating random numbers from negative binomial distribution
n_samples <- 1000
size <- 2
prob <- 0.3
random_numbers <- rnbinom(n_samples, size = size, prob = prob)

  7. Parameter Estimation for Negative Binomial Distribution in R:

    Estimate parameters from data using the fitdistr function.

    # Parameter estimation for negative binomial distribution
library(MASS)
fit_params <- fitdistr(random_numbers, densfun = "nbinom")

  8. Comparing Negative Binomial Density to Other Distributions in R:

    Compare negative binomial density to other distributions.

    # Comparing negative binomial density to other distributions
x_values <- 0:20
neg_binom_pmf <- dnbinom(x_values, size = size, prob = prob)
poisson_pmf <- dpois(x_values, lambda = size * (1 - prob) / prob)
geom_pmf <- dgeom(x_values, prob = prob)
plot(x_values, neg_binom_pmf, type = "h", lwd = 2, col = "blue", main = "Comparison of Distributions")
lines(x_values, poisson_pmf, type = "h", lwd = 2, col = "red")
lines(x_values, geom_pmf, type = "h", lwd = 2, col = "green")
legend("topright", legend = c("Negative Binomial", "Poisson", "Geometric"), col = c("blue", "red", "green"), lwd = 2)

  9. Plotting Negative Binomial Distribution in R:

    Visualize the negative binomial distribution using the hist function.

    # Plotting negative binomial distribution
hist(random_numbers, prob = TRUE, col = "lightblue", main = "Negative Binomial Distribution")
curve(dnbinom(x, size = size, prob = prob), add = TRUE, col = "blue", lwd = 2)

  10. Applications of Negative Binomial Distribution in R:

    Apply the negative binomial distribution in real-world scenarios, such as modeling counts with over-dispersion.

    # Applications of negative binomial distribution
# Example: Modeling counts with over-dispersion
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