Geometric Distribution: A New Probability Distribution with Mean 1/p

The geometric distribution is a discrete probability distribution that models the number of trials until the first success in a sequence of independent and identically distributed Bernoulli trials. In this article, we will explore the geometric distribution, its properties, and how to implement it using R.

Introduction to Geometric Distribution

The geometric distribution is commonly used to model situations where we have multiple attempts or trials to achieve a certain outcome. For example, in a manufacturing process, the number of attempts until the first successful product is a common application of this distribution.

The probability mass function (PMF) of the geometric distribution is given by:

p(x | p) = (1-p)^x * p

where x is the number of trials until the first success, and p is the probability of success on each trial.

Properties of Geometric Distribution

The geometric distribution has several properties that make it useful in modeling real-world phenomena. Some of these properties include:

• Mean: The expected value of the geometric distribution is given by 1/p, where p is the probability of success.
• Variance: The variance of the geometric distribution is given by (1-p)/p^2.
• Skewness: The skewness of the geometric distribution is given by 3/(4(1-p)).

Geometric Distribution in R

In R, the geometric distribution can be accessed through the rgeom() function. This function generates a random variable following a geometric distribution with specified parameters.

However, we need to adjust the formula slightly to get the desired distribution. The rgeom() function returns p(1-p)^(x), but we want p(1-p)^(x-1). To achieve this, we can use the following approach:

my_rgeom <- function(n, prob) rgeom(n, prob) + 1 - (1 - rgeom(n, prob))

This function generates a random variable following a geometric distribution with specified parameters and then subtracts 1 to adjust for the first success.

Geometric Distribution in R: Advanced Implementation

In addition to the basic rgeom() function, we can also implement more advanced functions using the geometric distribution. One such function is my_dgeom(), which calculates the cumulative distribution function (CDF) of the geometric distribution.

The CDF of a geometric distribution gives us the probability that x successes occur in n trials or fewer. We can calculate this using the following formula:

p(x | p, n) = 1 - (1-p)^n

To implement this function in R, we can use the following code:

my_dgeom <- function(x, prob, log = FALSE) {
  p <- dgeom(x, prob)
  if (isTRUE(log)) log(p) else p
}

This function calculates the CDF of the geometric distribution using the specified parameters and returns either the raw value or its logarithm.

Geometric Distribution in R: Applications

The geometric distribution has several applications in real-world scenarios. Some examples include:

• Manufacturing processes: The number of attempts until the first successful product is a common application of this distribution.
• Telecommunications: The number of calls until the first dropped call can be modeled using the geometric distribution.
• Finance: The number of trials until the first profitable transaction can also be modeled using this distribution.

Conclusion

In conclusion, the geometric distribution is a powerful tool for modeling real-world phenomena. We have explored its properties, implementation in R, and several applications. By understanding the geometric distribution, we can better model and analyze complex systems.

Additional Resources

If you are interested in learning more about the geometric distribution or other probability distributions, here are some additional resources:

• R documentation: The official R documentation has an extensive section on probability distributions, including the geometric distribution.
• Wikipedia: The Wikipedia article on the geometric distribution provides a comprehensive overview of its properties and applications.
• Coursera course: Coursera offers a course on probability theory that covers the geometric distribution and other related topics.

Last modified on 2023-11-24