Poisson & Exponential

Medium3–4 hrsPoissonExponentialMemorylessArrival Process

Overview

Poisson and Exponential are the same process viewed differently: Poisson counts events in a window; Exponential measures time between events. They're linked: if arrivals are Poisson(λ), inter-arrival times are Exp(λ). The memoryless property is the key feature: the expected future wait time doesn't depend on how long you've already waited. This makes Exponential the continuous analogue of Geometric.

How to Recognize

→Counting rare events in fixed time/space: 'calls per hour', 'trades per second'
→Waiting time between events: 'time until next trade', 'time until first failure'
→Keyword: 'arrives at rate λ', 'Poisson process', 'on average λ per unit time'

Step-by-Step Approach

1.Poisson(λ): E[X]=Var[X]=λ. Use when counting rare independent events
2.Exponential(λ): waiting time between Poisson events. E[T]=1/λ, memoryless
3.Min of independent Exp(λᵢ): also Exponential with rate Σλᵢ
4.Large n, small p, np=λ → Binomial ≈ Poisson

Key Formulas

P(X=k) = \frac{e^{-\lambda}\lambda^k}{k!},\quad E[X]=\operatorname{Var}[X]=\lambda

P(T > t) = e^{-\lambda t}

for

T\sim\operatorname{Exp}(\lambda)

E[\min(T_1,T_2)] = \frac{1}{\lambda_1+\lambda_2}

(independent exponentials)

04Memoryless:

P(T>s+t\mid T>s) = P(T>t) = e^{-\lambda t}

Worked Examples

0Quick Example

Orders arrive at rate λ=6/hr. P(wait>10min) = P(T > 1/6 hr) = e^(−6·1/6) = e^(−1) ≈ 36.8%.

Problem

Two types of events arrive: Type A at rate 3/hour, Type B at rate 2/hour. What is the expected time until ANY event? What is P(first event is Type A)?

Solution

Min of independent Exp(λ₁=3) and Exp(λ₂=2): also Exponential with rate λ₁+λ₂=5.

E[min] = 1/5 hour = 12 minutes.

P(first event is Type A) = λ_A/(λ_A+λ_B) = 3/5 = 60%.

This is the beautiful competing exponentials result: each 'race' to fire first, and the winner's probability equals its share of the total rate.

Answer

$E[\text{first event}] = 1/5$ hr = 12 min. $P(\text{Type A wins}) = 3/(3+2) = 60\%$ . Exponentials compete in proportion to their rates.

Common Mistakes

!
Applying memoryless property without checking: only Exponential and Geometric are memoryless.
!
Poisson mean = variance. If a problem says mean ≠ variance for count data, it's not Poisson.
!
For min of exponentials: the rate ADDS, so the minimum is faster. E[min] = 1/(Σλᵢ) < any individual 1/λᵢ.

Practice Problems

Click "Show Answer" to reveal

Emails arrive at rate 10/hour. What is P(no email in the next 30 minutes)?

Hint: P(T > 0.5 hr) = e^(−λt) = e^(−10·0.5).

3 taxi companies: A arrives at rate 2/hr, B at 3/hr, C at 5/hr. You call all three simultaneously. What is E[waiting time] and P(company A arrives first)?

Hint: Total rate = 2+3+5 = 10. P(A first) = 2/10.

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Normal Distribution & CLT Hypothesis Testing & p-values