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.