Probability Distribution: Exponential
The exponential distribution is a continuous probability distribution used to model the time between events in a Poisson process (i.e., events that occur independently and at a constant average rate).
Probability Density Function (PDF)
The Probability Density Function (PDF) gives the probability of the time between events.
\[f(x) = \lambda e^{-\lambda x}\]Where:
- x is the random variable representing the time between events ($x \ge 0$).
- $\lambda$ (lambda) is the rate parameter, representing the average number of events per unit of time.
- e is Euler’s number.
Derivations
Derivation of the Mean (E[X])
The mean, or expected time between events, is calculated by integrating $x \cdot f(x)$ over its entire range. This requires integration by parts ($\int u \,dv = uv - \int v \,du$).
Set up the integral:
\[E[X] = \int_{0}^{\infty} x \cdot \lambda e^{-\lambda x} \,dx\]Apply integration by parts: Let $u = x$ (so $du = dx$) and $dv = \lambda e^{-\lambda x} dx$ (so $v = -e^{-\lambda x}$).
\[E[X] = \left[-x e^{-\lambda x}\right]_{0}^{\infty} - \int_{0}^{\infty} (-e^{-\lambda x}) \,dx\]Evaluate the terms: The first term, $[-x e^{-\lambda x}]$, evaluates to 0 at both infinity (as $e^{-\lambda x}$ goes to zero faster than $x$ goes to infinity) and at 0. The second term becomes:
\[E[X] = \int_{0}^{\infty} e^{-\lambda x} \,dx = \left[-\frac{1}{\lambda} e^{-\lambda x}\right]_{0}^{\infty}\]This evaluates to $0 - (-\frac{1}{\lambda}) = \frac{1}{\lambda}$.
Final Result:
\[E[X] = \frac{1}{\lambda}\]
Derivation of the Variance (Var(X))
The variance is calculated using the formula $Var(X) = E[X^2] - (E[X])^2$.
Calculate E[X²]:
\[E[X^2] = \int_{0}^{\infty} x^2 \cdot \lambda e^{-\lambda x} \,dx\]We use integration by parts again. Let $u = x^2$ ($du = 2x \,dx$) and $dv = \lambda e^{-\lambda x} dx$ ($v = -e^{-\lambda x}$).
\[E[X^2] = \left[-x^2 e^{-\lambda x}\right]_{0}^{\infty} - \int_{0}^{\infty} (-e^{-\lambda x}) \cdot 2x \,dx\]The first term is 0. The second term simplifies to:
\[E[X^2] = \int_{0}^{\infty} 2x e^{-\lambda x} \,dx = \frac{2}{\lambda} \int_{0}^{\infty} x \lambda e^{-\lambda x} \,dx\]We recognize the integral as the definition of $E[X]$, which we already found to be $\frac{1}{\lambda}$.
\[E[X^2] = \frac{2}{\lambda} \cdot E[X] = \frac{2}{\lambda} \cdot \frac{1}{\lambda} = \frac{2}{\lambda^2}\]Calculate the Variance:
\[Var(X) = E[X^2] - (E[X])^2 = \frac{2}{\lambda^2} - \left(\frac{1}{\lambda}\right)^2 = \frac{2}{\lambda^2} - \frac{1}{\lambda^2}\]Final Result:
\[Var(X) = \frac{1}{\lambda^2}\]
Practical Examples
Customer Service: The exponential distribution can model the time between consecutive customer arrivals at a call center. If a center receives an average of 10 calls per hour ($\lambda=10$), the average time between calls would be $1/\lambda = 1/10$ of an hour, or 6 minutes. This helps in staffing and resource allocation.
Reliability Engineering: It’s used to model the lifetime of electronic components, like an LED light bulb, that fail at a constant rate rather than wearing out. If a type of bulb has a failure rate of 1 per 50,000 hours ($\lambda = 1/50000$), its mean time to failure (MTTF) is $1/\lambda = 50,000$ hours. This is crucial for setting warranties and maintenance schedules.
For the exponential distribution, P(a ≤ X ≤ b) indicates the probability that the time until the next event falls between time a and time b.
Parallel to the Poisson Distribution
The relationship is about timing vs. counting.
Exponential Distribution (Timing): $P(a ≤ X ≤ b)$ answers the question, “What is the chance the next phone call arrives between 2 and 5 minutes from now?” It’s about the timing of a single event.
Poisson Distribution (Counting): The equivalent concept would be finding the probability that the number of events falls within a certain range. For example, “What is the chance we get between 2 and 5 phone calls in the next hour?” You would calculate this by summing the individual probabilities:
\[P(2 \le \text{Count} \le 5) = P(X=2) + P(X=3) + P(X=4) + P(X=5)\]
In short, the exponential distribution gives a probability over a continuous interval of time, while the Poisson distribution gives a probability over a discrete range of counts.