Poisson Distribution

Modeling the number of events occurring in a fixed interval of time or space.

The "Rare Events" Distribution

The Poisson Distribution is used to model the number of times an event occurs within a specified interval. The key assumptions are that events are independent, the average rate of events is constant, and two events cannot occur at the exact same instant.

In finance, it's particularly useful for modeling rare events. For example, a credit analyst might use it to model the number of defaults in a large portfolio of loans over a month, or a trader might use it to model the number of times a stock's price jumps by more than 5% in a single day.

Interactive Poisson Distribution

Adjust the rate parameter (λ) to see how the shape of the distribution changes. Notice how for large λ, the distribution starts to look like a normal distribution.

Average Rate (λ): 5.0

Mean (

\mu

): 5.00

Variance (

\sigma^2

): 5.00

Core Concepts

Probability Mass Function (PMF)

The PMF answers: 'What is the probability of observing exactly k events in an interval?'

P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}

$k$ is the number of occurrences of an event (0, 1, 2, ...).
$\lambda$ (lambda) is the average number of events per interval.
$e$ is Euler's number (approximately 2.71828).
The $k!$ (k factorial) term is crucial because it accounts for the fact that the events can happen in any order within the interval.

Mean (

\mu

): 5.00

Variance (

\sigma^2

): 5.00

Cumulative Distribution Function (CDF)

The CDF answers: 'What is the probability of observing k events or fewer?'

F(k) = P(X \le k) = \sum_{i=0}^{k} \frac{\lambda^i e^{-\lambda}}{i!}

The CDF accumulates the probabilities from the PMF. For example, the probability of observing 2 or fewer events, $P(X \le 2)$ , is the sum of the probabilities of observing 0, 1, and 2 events: $P(X=0) + P(X=1) + P(X=2)$ .

Mean (

\mu

): 5.00

Variance (

\sigma^2

): 5.00

Key Derivations

Deriving the Mean and Variance

The mean and variance of a Poisson distribution are most elegantly derived using its Moment-Generating Function (MGF).

Deriving the Expected Value (Mean)

Step 1: The Moment-Generating Function (MGF)

The MGF of a Poisson( $\lambda$ ) random variable is defined as $M_X(t) = E[e^{tX}]$ . It can be shown to be:

M_X(t) = e^{\lambda(e^t - 1)}

Step 2: Find the First Derivative

The mean, $E[X]$ , is the first derivative of the MGF evaluated at $t=0$ . We use the chain rule:

M_X'(t) = \frac{d}{dt} e^{\lambda(e^t - 1)} = e^{\lambda(e^t - 1)} \cdot \frac{d}{dt}(\lambda(e^t - 1))

= e^{\lambda(e^t - 1)} \cdot \lambda e^t

Step 3: Evaluate at t=0

Now, we substitute $t=0$ into the first derivative.

E[X] = M_X'(0) = e^{\lambda(e^0 - 1)} \cdot \lambda e^0

= e^{\lambda(1 - 1)} \cdot \lambda \cdot 1 = e^0 \cdot \lambda = \lambda

Final Mean Formula

E[X] = \lambda

Deriving the Variance

We use $Var(X) = E[X^2] - (E[X])^2$ . We need the second moment, $E[X^2]$ , which is the second derivative of the MGF at $t=0$ .

Step 1: Find the Second Derivative

We differentiate $M_X'(t)$ using the product rule:

M_X''(t) = \frac{d}{dt} \left[ (e^{\lambda(e^t - 1)}) \cdot (\lambda e^t) \right]

= (e^{\lambda(e^t - 1)} \cdot \lambda e^t) \cdot (\lambda e^t) + (e^{\lambda(e^t - 1)}) \cdot (\lambda e^t)

= \lambda^2 e^{2t} e^{\lambda(e^t - 1)} + \lambda e^t e^{\lambda(e^t - 1)}

Step 2: Evaluate at t=0

Substitute $t=0$ into the second derivative to find $E[X^2]$ .

E[X^2] = M_X''(0) = \lambda^2 e^0 e^{\lambda(e^0-1)} + \lambda e^0 e^{\lambda(e^0-1)}

= (\lambda^2 \cdot 1 \cdot e^0) + (\lambda \cdot 1 \cdot e^0) = \lambda^2 + \lambda

Step 3: Calculate the Variance

Now we use the variance formula:

Var(X) = E[X^2] - (E[X])^2 = (\lambda^2 + \lambda) - (\lambda)^2 = \lambda

Final Variance Formula

Var(X) = \lambda

Applications

Quantitative Finance: Operational Risk

Banks use the Poisson distribution to model operational risk, such as the number of fraudulent transactions or system failures per week. If a bank's system typically has $\lambda=2$ failures per week, and one week they experience 8 failures, they can use the PMF to calculate how unlikely that event was. This triggers an investigation to see if something has fundamentally changed in their system's stability.

Machine Learning: Feature Engineering

In analyzing customer behavior data (e.g., website visits), the number of times a user performs a certain action in a day (like 'add to cart') might follow a Poisson distribution. Understanding this can help in feature engineering. Instead of just using the raw count, a data scientist might create a feature like "Is this user's activity a rare event?" by calculating the Poisson probability $P(X \ge k)$ .