Geometric distribution

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The geometric distribution describes the number of independent Bernoulli trials until the first successful outcome occurs -- for example, the number of coin tosses until the first heads. The parameter «p» is the probability of success on any given trial.

Functions

Geometric(p)

The distribution function. Use this to define an uncertain variable that represents a number of events that occur until a success occurs, where «p» is the probability of success on each trial.

ProbGeometric(k, p)

Returns the probability that «k» or fewer events occur until a success is seen. This is the analytic discrete probability function. It returns

[math]\displaystyle{ p(k) = p (1-p)^{k-1} }[/math]

CumGeometric(k ,p)

The analytic cumulative probability function for a Geometric distribution. This is the probability that the actual number of events that occur until a success occurs is less than or equal to «k», given that at each trial there is a «p» probability of success. The value is equal to

[math]\displaystyle{ F(k) = 1 - (1-p)^k }[/math]

CumGeometricInv(q, p)

The inverse of the cumulative probability function for the Geometric(p) distribution, also know as the quantile function.

Computes the smallest number of independent Bernoulli trials, k, such that the probability of seeing at most k of failures before the first success is greater than or equal to «u», where the probability of success of each trial is «p».

Statistics

The Geometric distribution has the following statistics

Min = 1
Mean = 1/p
Mode = 1
Variance = [math]\displaystyle{ (1-p)/p^2 }[/math]
Skewness = [math]\displaystyle{ {{2-p}\over{\sqrt{1-p}}} }[/math]

Examples

An aspiring gymnast catches her jaeger (a release move on uneven bars) at practice 40% of the time. Her coach wants her to successfully catch at least one during practice 95% of the time (i.e., in 95% of her practices, she should catch at least one). How many repetitions should the coach insist on during each practice?

CumGeometricInv(95%, 40%) → 6

The actual success rate if she makes 6 attempts every practice should be

CumGeometric(6, 40%) → 95.33%

If you enter a lottery every day of the year, where each entry has odds of 1 chance in 1M of winning, what is the probability that you will win within one year? How about within 10 years?

CumGeometric(365, 1/1M) → 3.649e-004

CumGeometric(3653, 1/1M) → 3.646e-003

What is the probability of rolling doubles ten times in a row with a pair of fair dice? To encode this, treat a success as a non-double roll, which has a probability of 5/6, so the answer is given by

1 - CumGeometric(10, 5/6) → 1.654e-008

John is a baseball player, who hits the ball on 10% of his swings. What is the probability he gets a hit during his next three swings?

CumGeometric(3, 10%)

History

The analytic functions (ProbGeometric, CumGeometric, and CumGeometricInv) were included as built-in functions in Analytica 5.2.

The analytic functions were provided in the Distribution Densities Library for the first time in the 4.4.3 patch release.