Bernoulli Distribution in R

The Bernoulli distribution is a probability distribution that describes the outcome of a single binary experiment, where the result can be one of two possible outcomes, typically labeled as success or failure. It is named after Swiss mathematician Jacob Bernoulli.

The Bernoulli distribution is defined by a single parameter, usually denoted by p, which represents the probability of success.

The probability of failure, q, is simply 1 – p.

The formula for the Bernoulli distribution is as follows:

P(X = 1) = p

P(X = 0) = 1 – p

where X is the random variable that takes on the value 1 for success and 0 for failure.

In other words, the probability of success is p and the probability of failure is 1-p. The expected value (mean) of the Bernoulli distribution is p, and the variance is p(1-p).

The Bernoulli distribution is a probability distribution that represents the probability of a binary outcome, such as the flip of a coin or the success or failure of an event. In R, there are several functions that can be used to calculate the Bernoulli distribution.

The probability mass function (PMF) of the Bernoulli distribution is defined as:

P(X = x) = p^x * (1-p)^(1-x) for x in {0,1}

In R, you can use the following functions to work with the Bernoulli distribution:

1. dbinom(x, size, prob) in R

dbinom(x, size, prob): computes the probability mass function (PMF) of the Bernoulli distribution at x, where size is the number of trials and prob is the probability of success.

Example: Compute the probability of getting exactly 2 heads in 5 tosses of a fair coin (p = 0.5) using the Bernoulli distribution.

x <- 2

n <- 5

p <- 0.5

dbinom(x, n, p)

Output:

[1] 0.3125

Here is another example code that plots the PMF for p = 0.2, p = 0.5, and p = 0.8:

probs <- c(0.2, 0.5, 0.8) # set the probabilities of success

x <- 0:1 # set the possible number of successes

# calculate the PMF for each probability of success

pmf <- sapply(probs, function(p)

dbinom(x, 1, p))

# create the plot

plot(

x,

pmf[, 1],

type = "h",

lwd = 2,

ylim = c(0, 1),

xlab = "Number of successes",

ylab = "Probability",

main = "Bernoulli Distribution PMF"

)

lines(x,

pmf[, 2],

type = "h",

lwd = 2,

col = "blue")

lines(x,

pmf[, 3],

type = "h",

lwd = 2,

col = "red")

legend(

"topright",

legend = probs,

col = c("black", "blue", "red"),

lwd = 2,

title = "Probability of success"

)

2. pbinom(q, size, prob) in R

 pbinom(q, size, prob): computes the cumulative distribution function (CDF) of the Bernoulli distribution at q.

Example: Compute the probability of getting 2 or fewer heads in 5 tosses of a fair coin (p = 0.5) using the Bernoulli distribution.

q <- 2

n <- 5

p <- 0.5

pbinom(q, n, p)

Output:

[1] 0.6875

Here’s another example code for using the pbinom() function to plot the cumulative distribution function (CDF) of a Bernoulli distribution:

# Set the probability of success

p <- 0.3

# Set the possible number of successes

x <- 0:1

# Calculate the CDF for each possible number of successes

cdf <- sapply(x, function(k)

pbinom(k, 1, p))

# Create the plot

plot(

x,

cdf,

type = "s",

lwd = 2,

ylim = c(0, 1),

xlab = "Number of successes",

ylab = "Cumulative Probability",

main = "Bernoulli Distribution CDF"

)

points(x, cdf, pch = 19)

segments(x, 0, x, cdf, lty = 2)

3. qbinom(p, size, prob) in R

qbinom(p, size, prob): computes the quantile function of the Bernoulli distribution at probability p.

Example: Find the number of trials required to get at least 2 heads with probability 0.7 using the Bernoulli distribution (p = 0.5).

p <- 0.7

n <- qbinom(p, 100, 0.5)

n

Output:

[1] 3

Here’s an example of how to use qbinom and plot the results:

# set the probability of success and number of trials

p <- 0.3

n <- 10

# calculate the 0.05, 0.5, and 0.95 quantiles of the binomial distribution

q <- qbinom(c(0.05, 0.5, 0.95), size = n, prob = p)

# plot the cumulative distribution function (CDF) of the binomial distribution

x <- 0:n

y <- pbinom(x, size = n, prob = p)

plot(x,

y,

type = "s",

xlab = "Number of successes",

ylab = "Cumulative probability")

# add vertical lines for the quantiles

abline(

v = q,

col = c("red", "blue", "green"),

lty = c(1, 2, 3)

)

4. rbinom(n, size, prob) in R

rbinom(n, size, prob): generates n random values from a Bernoulli distribution with parameter p.

Example: Generate 100 random values from a Bernoulli distribution with probability of success (p) 0.3.

# Generate 100 random values from 

# a Bernoulli distribution with p = 0.3

set.seed(123) # Setting seed for reproducibility

n <- 100 # Sample size

p <- 0.3 # Probability of success

x <- rbinom(n, 1, p) 

# Generate 100 random values from a Bernoulli distribution

x

Output:

[1] 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 

0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 

0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Here’s another example of how to use rbinom and plot the results in R:

# set the probability of success and number of trials

p <- 0.3

n <- 10

# generate 1000 random samples from the binomial distribution

samples <- rbinom(1000, size = n, prob = p)

# plot a histogram of the samples

hist(

samples,

breaks = seq(-0.5, n + 0.5, by = 1),

freq = FALSE,

main = "Histogram of Binomial Samples",

xlab = "Number of Successes"

)

# overlay the theoretical probability mass function (PMF)

x <- 0:n

pmf <- dbinom(x, size = n, prob = p)

lines(x,

pmf,

type = "h",

lwd = 2,col = "red")

Binomial Distribution in R Programming

Bernoulli distribution vs Binomial distribution

The Bernoulli distribution and the binomial distribution are related probability distributions, but they have different characteristics and applications.

The Bernoulli distribution is a discrete probability distribution that describes a single experiment or trial with two possible outcomes – success or failure – and a fixed probability of success. The parameter of the Bernoulli distribution is the probability of success, denoted by p.

The binomial distribution, on the other hand, describes the probability of obtaining a certain number of successes in a fixed number of independent Bernoulli trials. It is a discrete probability distribution that has two parameters – the number of trials, denoted by n, and the probability of success on each trial, denoted by p.

To summarize:

• The Bernoulli distribution describes a single trial with two possible outcomes – success or failure.

• The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials.

• The Bernoulli distribution is a special case of the binomial distribution when the number of trials is 1.

• The binomial distribution can be thought of as a collection of Bernoulli trials.

In practical terms, the Bernoulli distribution is often used to model simple yes/no events, such as the flipping of a coin, while the binomial distribution is used to model more complex events where the probability of success may vary but the number of trials is fixed, such as the number of heads obtained when flipping a coin multiple times.

Binomial distribution

In R programming, you can use the built-in dbinom(), pbinom(), qbinom(), and rbinom() functions to work with the binomial distribution.

Here’s a brief explanation of each function:

1. dbinom(x, size, prob) calculates the probability mass function (PMF) of the binomial distribution at a specific value of x. size is the number of trials and prob is the probability of success on each trial.

2. pbinom(q, size, prob) calculates the cumulative distribution function (CDF) of the binomial distribution up to a specific quantile q.

3. qbinom(p, size, prob) calculates the inverse CDF of the binomial distribution, which gives the quantile q such that the probability of observing a value less than or equal to q is p.

4. rbinom(n, size, prob) generates n random samples from the binomial distribution with parameters size and prob.

Here’s an example usage of these functions:

# Calculate the PMF of the binomial distribution 

# at x = 3, with size = 10 and prob = 0.5

dbinom(3, size = 10, prob = 0.5)

# Calculate the CDF of the binomial distribution 

# up to q = 5, with size = 10 and prob = 0.5

pbinom(5, size = 10, prob = 0.5)

# Calculate the quantile q such that the probability of 

# observing a value less than or equal to q is p = 0.3,

# with size = 10 and prob = 0.5

qbinom(0.3, size = 10, prob = 0.5)

# Generate 10 random samples from the binomial distribution 

# with size = 10 and prob = 0.5

rbinom(10, size = 10, prob = 0.5)

These functions are useful for performing various statistical analyses and simulations involving the binomial distribution in R.

Bernoulli distribution

The binomial distribution and Bernoulli distribution are closely related, but there are some important differences in how they are used in R.

The Bernoulli distribution is a special case of the binomial distribution where there is only one trial (i.e., n=1). In the Bernoulli distribution, the outcome of the trial is either success or failure, with a probability of success denoted by p.

In R, there are two main functions to work with the Bernoulli distribution:

1. dbinom(x, 1, prob) – returns the probability mass function (pmf) for the Bernoulli distribution. It calculates the probability of x successes in one trial with a probability of success of prob.

2. rbinom(n, 1, prob) – generates n random numbers from a Bernoulli distribution. Each number represents the outcome of a single trial, with a probability of success of prob.

Here’s an example of how to use these functions in R:

# Probability mass function for Bernoulli distribution

dbinom(1, 1, 0.5) 

# Probability of success in one trial with p=0.5

# Random number generation from Bernoulli distribution

rbinom(5, 1, 0.5) 

# Generate 5 random outcomes from a 

# Bernoulli distribution with p=0.5

As you can see, the main difference between the binomial and Bernoulli distribution in R is the value of n (the number of trials). In the Bernoulli distribution, n=1, whereas in the binomial distribution, n can be any positive integer.

Poisson Functions in R Programming

Poisson distribution

Poisson distribution is a discrete probability distribution that describes the number of independent events that occur in a fixed interval of time or space. It is named after the French mathematician Siméon Denis Poisson, who introduced it in the early 19th century.

The Poisson distribution is characterized by a single parameter λ (lambda), which represents the average rate at which events occur. The probability mass function (PMF) of the Poisson distribution is given by:

P(X = k) = e^(-λ) * λ^k / k!

where k is a non-negative integer and k! is the factorial of k.

The mean and variance of the Poisson distribution are both equal to λ. This means that the distribution becomes more concentrated around the mean as λ increases.

The Poisson distribution is often used to model rare events such as accidents, defects, or occurrences of a specific disease in a population. It is also used in queueing theory, reliability analysis, and other fields where the occurrence of discrete events is of interest.

In summary, the Poisson distribution is a probability distribution that models the number of independent events occurring within a fixed interval of time or space, where the mean and variance of the distribution are equal to λ, the average rate of occurrence.

In R programming, there are several built-in functions that can be used to work with Poisson distributions. Here are some commonly used Poisson functions in R:

1. dpois() function in R

dpois() function is used to calculate the probability mass function (PMF) for a given Poisson distribution. It takes two arguments: the value at which to evaluate the PMF and the mean of the Poisson distribution.

For example, to calculate the PMF at x = 3 for a Poisson distribution with a mean of 2.5, you can use the following code:

dpois(3, 2.5)

# Plot PMF for Poisson distribution with 

# mean of 2.5 for x = 0 to 10

x <- seq(0, 10, by = 1)

pmf <- dpois(x, 2.5)

plot(

x,

pmf,

type = "h",

lwd = 3,

main = "Poisson PMF",

xlab = "Number of events",

ylab = "Probability"

)

This will create a plot of the PMF for the Poisson distribution with a mean of 2.5, showing the probability of observing each number of events from 0 to 10. The type = "h" argument specifies that a histogram-style plot should be used, and lwd = 3 sets the line width to 3 to make the plot more visible. The main, xlab, and ylab arguments specify the plot title, x-axis label, and y-axis label, respectively.

2. ppois() function in R

ppois() function is used to calculate the cumulative distribution function (CDF) for a Poisson distribution. It takes two arguments: the value at which to evaluate the CDF and the mean of the Poisson distribution.

For example, to calculate the CDF at x = 3 for a Poisson distribution with a mean of 2.5, you can use the following code:

ppois(3, 2.5)

# Plot CDF for Poisson distribution with mean of 2.5 for q = 0 to 10

q <- seq(0, 10, by = 1)

cdf <- ppois(q, 2.5)

plot(

q,

cdf,

type = "s",

lwd = 3,

main = "Poisson CDF",

xlab = "Number of events",

ylab = "Probability"

)

This will create a plot of the CDF for the Poisson distribution with a mean of 2.5, showing the probability of observing up to each number of events from 0 to 10. The type = "s" argument specifies that a step function plot should be used, and lwd = 3 sets the line width to 3 to make the plot more visible. The main, xlab, and ylab arguments specify the plot title, x-axis label, and y-axis label, respectively.

3. qpois() function in R

qpois() function is used to calculate the quantiles of a Poisson distribution. It takes two arguments: the probability at which to evaluate the quantile and the mean of the Poisson distribution.

For example, to calculate the 95th percentile of a Poisson distribution with a mean of 2.5, you can use the following code:

qpois(0.95, 2.5)

# Plot ICDF for Poisson distribution 

# with mean of 2.5 for p = 0.1 to 0.9

p <- seq(0.1, 0.9, by = 0.1)

icdf <- qpois(p, 2.5)

plot(

p,

icdf,

type = "o",

pch = 19,

lwd = 3,

main = "Poisson ICDF",

xlab = "Probability",

ylab = "Number of events"

)

This will create a plot of the ICDF for the Poisson distribution with a mean of 2.5, showing the smallest integer value of x for which the probability of observing up to x events is greater than or equal to each probability value from 0.1 to 0.9. The type = "o" argument specifies that a line and points plot should be used, and pch = 19 sets the point character to a solid circle. The lwd = 3 argument sets the line width to 3 to make the plot more visible. The main, xlab, and ylab arguments specify the plot title, x-axis label, and y-axis label, respectively.

4. qpois() function in R

rpois() function is used to generate random numbers from a Poisson distribution. It takes one argument: the mean of the Poisson distribution.

For example, to generate 10 random numbers from a Poisson distribution with a mean of 2.5, you can use the following code:

rpois(10, 2.5)

# Define the values for x

x <- 0:10

# Compute the Poisson probability mass function for mean 3

pmf <- dpois(x, lambda = 3)

# Plot the probability mass function

plot(

x,

pmf,

type = "h",

lwd = 2,

col = "blue",

xlab = "x",

ylab = "P(X = x)",

main = "Poisson PMF for lambda = 3"

)

# Add text labels to the plot

text(6, pmf[7], "P(X=6) = 0.050")

# Add vertical and horizontal lines to highlight the label

segments(6, 0, 6, pmf[7], lty = 2)

segments(0, pmf[7], 6, pmf[7], lty = 2)

This will create a plot that shows the Poisson probability mass function for a Poisson distribution with mean 3, as well as a label indicating the probability of observing x = 6.

These functions can be used to perform a variety of calculations and simulations involving Poisson distributions in R programming.