Geometric distribution
In probability theory and statistics, the geometric distribution is either of two discrete probability distributions:
- The probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set
- The probability distribution of the number Y = X − 1 of failures before the first success, supported on the set
These two different geometric distributions should not be confused with each other. Often, the name shifted geometric distribution is adopted for the former one ; however, to avoid ambiguity, it is considered wise to indicate which is intended, by mentioning the support explicitly.
The geometric distribution gives the probability that the first occurrence of success requires k independent trials, each with success probability p. If the probability of success on each trial is p, then the probability that the kth trial is the first success is
for k = 1, 2, 3,....
The above form of the geometric distribution is used for modeling the number of trials up to and including the first success. By contrast, the following form of the geometric distribution is used for modeling the number of failures until the first success:
for k = 0, 1, 2, 3, ....
In either case, the sequence of probabilities is a geometric sequence.
For example, suppose an ordinary die is thrown repeatedly until the first time a "1" appears. The probability distribution of the number of times it is thrown is supported on the infinite set and is a geometric distribution with p = 1/6.
The geometric distribution is denoted by Geo where 0 < p ≤ 1.
Definitions
Consider a sequence of trials, where each trial has only two possible outcomes. The probability of success is assumed to be the same for each trial. In such a sequence of trials, the geometric distribution is useful to model the number of failures before the first success. The distribution gives the probability that there are zero failures before the first success, one failure before the first success, two failures before the first success, and so on.Assumptions: When is the geometric distribution an appropriate model?
The geometric distribution is an appropriate model if the following assumptions are true.- The phenomenon being modeled is a sequence of independent trials.
- There are only two possible outcomes for each trial, often designated success or failure.
- The probability of success, p, is the same for every trial.
An alternative formulation is that the geometric random variable X is the total number of trials up to and including the first success, and the number of failures is X − 1. In the graphs above, this formulation is shown on the left.
Probability Outcomes Examples
The general formula to calculate the probability of k failures before the first success, where the probability of success is p and the probability of failure is q = 1 − p, isfor k = 0, 1, 2, 3,....
E1) A doctor is seeking an anti-depressant for a newly diagnosed patient. Suppose that, of the available anti-depressant drugs, the probability that any particular drug will be effective for a particular patient is p = 0.6. What is the probability that the first drug found to be effective for this patient is the first drug tried, the second drug tried, and so on? What is the expected number of drugs that will be tried to find one that is effective?
The probability that the first drug works. There are zero failures before the first success. Y = 0 failures. The probability P is simply the probability that the first drug works.
The probability that the first drug fails, but the second drug works. There is one failure before the first success. Y= 1 failure. The probability for this sequence of events is P p which is given by
The probability that the first drug fails, the second drug fails, but the third drug works. There are two failures before the first success. Y = 2 failures. The probability for this sequence of events is P p P
E2) A newlywed couple plans to have children, and will continue until the first girl. What is the probability that there are zero boys before the first girl, one boy before the first girl, two boys before the first girl, and so on?
The probability of having a girl is p= 0.5 and the probability of having a boy is q = 1 − p = 0.5.
The probability of no boys before the first girl is
The probability of one boy before the first girl is
The probability of two boys before the first girl is
and so on.
Properties
Moments and cumulants
The expected value for the number of independent trials to get the first success, of a geometrically distributed random variable X is 1/p and the variance is /p2:Similarly, the expected value of the geometrically distributed random variable Y = X − 1 is q/p = /p, and its variance is /p2:
Let μ = /p be the expected value of Y. Then the cumulants of the probability distribution of Y satisfy the recursion
Outline of proof: That the expected value is /p can be shown in the following way. Let Y be as above. Then
Expected Value Examples
E3) A patient is waiting for a suitable matching kidney donor for a transplant. If the probability that a randomly selected donor is a suitable match is p=0.1, what is the expected number of donors who will be tested before a matching donor is found?With p = 0.1, the mean number of failures before the first success is E = /p =/0.1 = 9.
For the alternative formulation, where X is the number of trials up to and including the first success, the expected value is E = 1/p = 1/0.1 = 10.
For example 1 above, with p = 0.6, the mean number of failures before the first success is E = /p = /0.6 = 0.67.
General properties
- The probability-generating functions of X and Y are, respectively,
- Like its continuous analogue, the geometric distribution is memoryless. That means that if you intend to repeat an experiment until the first success, then, given that the first success has not yet occurred, the conditional probability distribution of the number of additional trials does not depend on how many failures have been observed. The die one throws or the coin one tosses does not have a "memory" of these failures. The geometric distribution is the only memoryless discrete distribution.
- Among all discrete probability distributions supported on with given expected value μ, the geometric distribution X with parameter p = 1/μ is the one with the largest entropy.
- The geometric distribution of the number Y of failures before the first success is infinitely divisible, i.e., for any positive integer n, there exist independent identically distributed random variables Y1, ..., Yn whose sum has the same distribution that Y has. These will not be geometrically distributed unless n = 1; they follow a negative binomial distribution.
- The decimal digits of the geometrically distributed random variable Y are a sequence of independent random variables. For example, the hundreds digit D has this probability distribution:
- Golomb coding is the optimal prefix code for the geometric discrete distribution.
- The sum of two independent Geo distributed random variables is not a geometric distribution.
Related distributions
- The geometric distribution Y is a special case of the negative binomial distribution, with r = 1. More generally, if Y1, ..., Yr are independent geometrically distributed variables with parameter p, then the sum
- The geometric distribution is a special case of discrete compound Poisson distribution.
- If Y1, ..., Yr are independent geometrically distributed variables, then their minimum
- Suppose 0 < r < 1, and for k = 1, 2, 3, ... the random variable Xk has a Poisson distribution with expected value r k/k. Then
- The exponential distribution is the continuous analogue of the geometric distribution. If X is an exponentially distributed random variable with parameter λ, then
- If p = 1/n and X is geometrically distributed with parameter p, then the distribution of X/n approaches an exponential distribution with expected value 1 as n → ∞, since
Statistical Inference
Parameter estimation
For both variants of the geometric distribution, the parameter p can be estimated by equating the expected value with the sample mean. This is the method of moments, which in this case happens to yield maximum likelihood estimates of p.Specifically, for the first variant let k = k1, ..., kn be a sample where ki ≥ 1 for i = 1, ..., n. Then p can be estimated as
In Bayesian inference, the Beta distribution is the conjugate prior distribution for the parameter p. If this parameter is given a Beta prior, then the posterior distribution is
The posterior mean E approaches the maximum likelihood estimate as α and β approach zero.
In the alternative case, let k1, ..., kn be a sample where ki ≥ 0 for i = 1, ..., n. Then p can be estimated as
The posterior distribution of p given a Beta prior is
Again the posterior mean E approaches the maximum likelihood estimate as α and β approach zero.
For either estimate of using Maximum Likelihood, the bias is equal to
which yields the bias-corrected maximum likelihood estimator
Computational methods
Geometric distribution using R
The R function dgeom
calculates the probability that there are k failures before the first success, where the argument "prob" is the probability of success on each trial.For example,
dgeom = 0.6
dgeom = 0.24
R uses the convention that k is the number of failures, so that the number of trials up to and including the first success is k + 1.
The following R code creates a graph of the geometric distribution from Y = 0 to 10, with p = 0.6.
Y=0:10
plot, type="h", ylim=c, main="Geometric distribution for p=0.6", ylab="P
Geometric distribution using Excel
The geometric distribution, for the number of failures before the first success, is a special case of the negative binomial distribution, for the number of failures before s successes.The Excel function
NEGBINOMDIST
calculates the probability of k = number_f failures before s = number_s successes where p = probability_s is the probability of success on each trial. For the geometric distribution, let number_s = 1 success.For example,
=NEGBINOMDIST
= 0.6=NEGBINOMDIST
= 0.24Like R, Excel uses the convention that k is the number of failures, so that the number of trials up to and including the first success is k + 1.