Geometric distribution | Vose Software

# Geometric distribution

Format: Geometric(p)

Geometric(p) models the number of failures that will occur before the first success in a set of binomial trials, given that p is the probability of a trial succeeding. Examples of the Geometric distribution are shown below:

## Examples

### Cards (for simplicity)

I select a card from a pack (no jokers) and guess its suit before looking at it. The number of wrong guesses I will have before I get it right can be estimated as Geometric(25%). The Geometric distribution assumes that p is constant with each trial i.e. that I cannot get any better at guessing with each failure, nor does my problem change (so I'll have to put the card back and reshuffle). It also assumes that I will doggedly carry on, even if it takes me a hundred failures before I succeed. Thus, some caution is needed in its application.

### Survey example

A company wants to do a random survey of past purchasers of its product to find one who has experienced a particular problem you know exists with their product, and then look at the damage this fault produced. They know that 1.5% of their products sold will likely have this problem. The number of purchasers they'll need to contact is given by 1 + Geometric(1.5%).

### Dry oil wells

The Geometric distribution is sometimes quoted as useful to estimate the number of dry wells an oil company will drill in a particular section before getting a producing well. That would, however, be assuming that a) the company doesn't learn from its mistakes; and b) it has the money and obstinacy to keep drilling new wells despite the cost.

The Geometric distribution is a special case of the Negative Binomial for s = 1 i.e. Geometric(p) = NegBin(1, p), which means that the sum of s independent Geometric(p) distributions = NegBin(s, p). The Geometric distribution is the discrete analogue of the Exponential distribution, and gets its name because its probability mass function is a geometric progression. The Geometric distribution is occasionally called a Furry distribution.

The Beta-Geometric distribution is an extension of the Geometric, where the probability p is itself a random variable.

## Zero-modified versions

When modeling or analyzing counting data, it is often desirable to modify the probability of zero of the discrete distribution we use, to more accurately model the probability of "no event occurring". We can make two types of modifications to our distribution for this:

• Zero-inflated model - we increase the probability of zero.

• Zero-truncated model - we entirely remove the probability of zero events occurring.

## ModelRisk functions added to Microsoft Excel for the Geometric distribution

VoseGeometric generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseGeometricObject constructs a distribution object for this distribution.

VoseGeometricProb returns the probability mass or cumulative distribution function for this distribution.

VoseGeometricProb10 returns the log10 of the probability mass or cumulative distribution function.

VoseGeometricFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseGeometricFitObject constructs a distribution object of this distribution fitted to data.

VoseGeometricFitP returns the parameters of this distribution fitted to data.

## ModelRisk functions added to Microsoft Excel for the Zero-Inflated Geometric distribution

VoseZIGeometric generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseZIGeometricObject constructs a distribution object for this distribution.

VoseZIGeometricProb returns the probability mass or cumulative distribution function for this distribution.

VoseZIGeometricProb10 returns the log10 of the probability mass or cumulative distribution function.

VoseZIGeometricFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseZIGeometricFitObject constructs a distribution object of this distribution fitted to data.

VoseZIGeometricFitP returns the parameters of this distribution fitted to data.

## ModelRisk functions added to Microsoft Excel for the Zero-Truncated Geometric distribution

VoseZTGeometric generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseZTGeometricObject constructs a distribution object for this distribution.

VoseZTGeometricProb returns the probability mass or cumulative distribution function for this distribution.

VoseZTGeometricProb10 returns the log10 of the probability mass or cumulative distribution function.

VoseZTGeometricFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseZTGeometricFitObject constructs a distribution object of this distribution fitted to data.

VoseZTGeometricFitP returns the parameters of this distribution fitted to data.

Geometric distribution equations

Zero-Inflated Geometric distribution equations

Zero-Truncated Geometric distribution equations