The Geometric Distribution

Number of Bernoulli trials until the first success, with the memoryless property

Introduction

The geometric distribution counts the number of Bernoulli trials required to obtain the first success. It is the discrete analogue of the exponential waiting-time distribution and shares the memoryless property: the probability of further trials to success does not depend on how many have already been tried.

Prerequisites

Bernoulli trials.

Theory

Two conventions exist. Let \(X\) be the number of Bernoulli\((p)\) trials up to and including the first success:

\[P(X = k) = (1 - p)^{k - 1} p, \qquad k = 1, 2, 3, \ldots\]

Alternatively, \(Y\) is the number of failures before the first success:

\[P(Y = k) = (1 - p)^k p, \qquad k = 0, 1, 2, \ldots\]

R uses the second convention (rgeom, dgeom return \(Y\), i.e., failures before first success).

Moments for the first convention (\(X\) = trials until first success):

• \(E[X] = 1/p\), \(\mathrm{Var}(X) = (1 - p)/p^2\).

For \(Y\) (R’s convention): \(E[Y] = (1 - p)/p\), \(\mathrm{Var}(Y) = (1 - p)/p^2\).

Memoryless property: \(P(X > m + n \mid X > m) = P(X > n)\). Having waited \(m\) trials without success does not change the distribution of further waiting time.

Assumptions

Independent Bernoulli trials with constant probability \(p\).

R Implementation

p <- 0.3

# R's convention: Y = failures before first success
Y <- rgeom(1e5, p)
c(mean_Y = mean(Y), theoretical = (1 - p) / p)

# Number of trials to first success X = Y + 1
X <- Y + 1
c(mean_X = mean(X), theoretical = 1 / p)

# PMF comparison
k <- 0:15
data.frame(k, pmf_Y = dgeom(k, p))

# Memoryless: P(Y > 5 + 3 | Y > 5) = P(Y > 3)
Y_past_5 <- Y[Y > 5]
c(conditional = mean(Y_past_5 > 5 + 3),
  unconditional = mean(Y > 3))

Output & Results

   mean_Y theoretical
    2.334       2.333

   mean_X theoretical
    3.334       3.333

    k      pmf_Y
1   0     0.3000
2   1     0.2100
3   2     0.1470
...

conditional  unconditional
     0.3441         0.3430

The memoryless property holds empirically: conditional probability of further 3 failures, given already 5 failures, equals the unconditional probability of at least 3 failures.

Interpretation

The geometric distribution is the baseline model for “time to event” in the discrete case: cycles to pregnancy in IVF, attempts before a successful procedure, trials before a rare positive test. Memorylessness is a strong assumption that often fails in real data where fatigue or learning changes \(p\) over time.

Practical Tips

• Check which convention (trials vs. failures) is expected when comparing to textbook formulas.
• Memorylessness is testable empirically: compare the distribution of residual trials given survival to each time.
• For non-memoryless discrete waiting times, the negative binomial (shape \(> 1\)) captures over-dispersion relative to geometric.
• The geometric converges to the exponential when discretising time with fine intervals.
• In clinical or biological settings, censoring is common; naive rgeom ignores right-censoring – use survival methods for realistic data.