The Exponential Distribution

Introduces the exponential distribution as a continuous probability model for waiting times. Covers the PDF $f(x)=\lambda e^{-\lambda x}$ , the CDF $F(x)=1-e^{-\lambda x}$ , computing tail and interval probabilities, and the relationship between the rate parameter and the mean.

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Tutorial

The Exponential PDF and CDF

A continuous random variable $X$ follows the exponential distribution with rate parameter $\lambda > 0$ if its probability density function is

f(x) = \begin{cases} \lambda e^{-\lambda x} & \text{if } x \geq 0, \\ 0 & \text{if } x < 0. \end{cases}

We write $X \sim \text{Exp}(\lambda)$ . The exponential distribution models waiting times: how long until the next phone call, the next radioactive decay, or the next equipment failure.

Integrating the PDF from $0$ to $x$ gives the cumulative distribution function:

F(x) = P(X \leq x) = 1 - e^{-\lambda x}, \quad x \geq 0.

The right-tail (or survival) probability is then

P(X > x) = 1 - F(x) = e^{-\lambda x}.

For example, if $X \sim \text{Exp}(2)$ , then

P(X > 1) = e^{-2 \cdot 1} = e^{-2} \approx 0.135.