Geometric Distribution Percent Point (Quantile)

Find the smallest failure count k where the cumulative probability P(X less or equal k) reaches q for a geometric distribution. Shows PMF, CDF, upper tail, mean, variance and bar charts.

Input

Enter probability q and success probability p to find the smallest failure count k (percent point) where the geometric cumulative probability P(X less or equal k) reaches q.

Probability q

Cumulative probability greater than 0 and at most 1. Example: 0.95

Success probability p

Value greater than 0 and at most 1. Example: 0.3

Result

Smallest failure count k for q = 0.95 and p = 0.3

Cumulative P(X less or equal k)

0.95964639

Probability mass P(X equals k)

0.0172944

Upper tail P(X greater or equal k)

0.05764801

Mean

2.33333333

Variance

7.77777778

Probability mass P(X equals k)

Cumulative distribution P(X less or equal k)

How it works

For the geometric distribution (failure count convention), the probability mass of the number of failures k before the first success is P(X=k)=(1-p)^k times p for k=0,1,2,..., where p is the success probability.
The cumulative distribution is P(X less or equal k)=1-(1-p)^(k+1). The percent point is the smallest integer k for which this is at least q.
A closed form is k=ceil(ln(1-q)/ln(1-p))-1. This tool adds a cumulative-probability check to guard against floating point error and guarantee the minimal k.
The mean is (1-p)/p and the variance is (1-p)/p^2. The upper tail probability is P(X greater or equal k)=1-P(X less or equal k-1).
When q is very close to 1 or the success probability p is very small, k can become very large.

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