• Used to model time between events or claim sizes
• Assumes memoryless property, ideal for small, frequent claims

In actuarial science, the exponential distribution is often used to model:

• Small insurance claims ( health, auto, and sometimes property)
• Time until the next claim
• Simplicity and memoryless property

\[ f(x; \lambda) = \lambda e^{-\lambda x}, \quad x > 0 \]

We’ll simulate 500 small claim amounts with a true mean of $1000.

Given \(n\) claim amounts \(x_1, x_2, ..., x_n\), the MLE for \(\lambda\) is:

\[ \hat{\lambda} = \frac{1}{\bar{x}} = \frac{n}{\sum x_i} \]

We can estimate the parameter from our sample.

## [1] 0.0009929488

The MLE of \(\lambda\) is approximately 9.9^{-4}.

• Exponential distribution is useful in modeling small, frequent insurance claims.
• MLE allows actuaries to estimate model parameters from data.
• Visualizing the data helps validate the model fit.