Exponential Distribution

This is the a special case of Gamma Distribution. In Gamma, we measure waiting period for $N$ events. In Exponential, we only measure waiting period for $1$ event.

If $k = 1$,

$$\text{Gamma}(1, \lambda) \equiv \text{Exponential}(\lambda)$$

1. The Core Idea

If events occur continuously and independently at a constant average rate $\lambda$ (a Poisson process), the waiting time $T$ until the next event occurs follows an Exponential distribution:

$$T \sim \text{Exponential}(\lambda)$$

Where:

• $\lambda$ (Lambda): The rate parameter (average number of events per unit of time).
• $T$: The continuous random variable representing time ($T \ge 0$).

PDF

$$f(t) = \lambda e^{-\lambda t} \quad (t \ge 0)$$

CDF

$$F(t) = P(T \le t) = 1 - e^{-\lambda t}$$

Properties

Mean (Expected Value)

$$E[T] = \frac{1}{\lambda}$$

Variance

$$\text{Var}(T) = \frac{1}{\lambda^2}$$

$\lambda$ - Rate (Events per unit time) $\beta=\frac{1}{\lambda}$ - Mean Scale (average time per event)

This distribution hold the Memorylessness of Distributions property.