Waiting-Time Distributions: Geometric and Poisson

"On average a customer files 3 claims a year." Is that the mean of a geometric count or a Poisson rate? The geometric counts trials until a success; the Poisson counts events in a fixed window. They share the actuary's favorite shortcut and one fatal trap each.

Repeat an independent trial with success probability $p$ until the first success. Two conventions appear, and the exam uses both.

Trials convention, $N$ = the trial on which the first success occurs:

$$P(N=n)=(1-p{)}^{n-1}p,n=1,2,3,\dots$$

$E[N]=1/p$, $\text{Var}(N)=(1-p)/p^2$. Failures convention, $Y=N-1$ = the number of failures before the first success: $E[Y]=(1-p)/p$, with the same variance. The survival function is clean:

$$P(N>n)=(1-p{)}^n$$

HIGH-FREQUENCY: The geometric is the only discrete memoryless distribution. Given no success in the first $k$ trials, the additional wait has the same distribution as a fresh start, so $E[N\mid N>k]=k+1/p$.

Common mistakes

• Mixing the two geometric conventions. $E=1/p$ counts trials; $E=(1-p)/p$ counts failures. A "mean of 5 visits" is ambiguous until you decide which. Read whether the count starts at 1 (trials) or 0 (failures).
• Forgetting the geometric is memoryless. $E[N\mid N>k]=k+1/p$, not $1/p$. Past failures do not shorten the remaining wait.
• Integrating the Poisson. It is discrete. $P(N\le 2)=P(0)+P(1)+P(2)$, summed, never integrated.

Bottom line

• Geometric($p$): trials until the first success. Trials $N\in \{1,2,\dots \}$: $E=1/p$, $\text{Var}=(1-p)/p^2$. Failures $Y=N-1$: $E=(1-p)/p$, same variance.
• Geometric survival: $P(N>n)=(1-p{)}^n$. It is memoryless: $E[N\mid N>k]=k+1/p$.
• Geometric variance exceeds its mean whenever $p<1$, since $(1-p)/p^2>1/p$; only the Poisson has $E=\text{Var}$.
• Poisson($\lambda$): events in a fixed window. $P(N=k)=e^{-\lambda }{\lambda }^k/k!$, and $E=\text{Var}=\lambda$.

Exam shortcut

Decide the geometric convention first (trials start at 1, failures at 0), then the mean follows. For a Poisson given a moment, write $E[N^2]=\lambda +{\lambda }^2$ and solve the quadratic. For combined Poisson streams, "add the lambdas, then compute one probability." "Geometric forgets, Poisson adds." Memorylessness for the geometric; rate-addition for independent Poissons.

The full lesson (about 1,705 words, 11 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 2a
• 2b
• 2c
• 2d
• 2e
• 2f

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