Exponential Distribution

Core Concepts

If a random variable $X$ follows an Exponential distribution, we write:

Where:

• $\lambda$ (lambda) is the rate parameter (the exact same rate parameter used in the Poisson distribution!).

Probability Density Function (PDF)

The probability density function creates a rapidly decaying curve:

Cumulative Distribution Function (CDF)

For continuous distributions, the CDF is often more useful than the PDF because it directly gives us the probability that the time between events will be less than or equal to $x$.

The "Memoryless" Property

This means the probability of an event occurring in the next $t$ minutes is completely independent of how long you have already been waiting.

Example: If the lifespan of a lightbulb is exponentially distributed, a lightbulb that has been working for 5 years is exactly as likely to break in the next 10 minutes as a brand new lightbulb.

Key Metrics

• Expected Value (Mean): $E[X] = \frac{1}{\lambda}$
  • (Intuition: If a bus arrives at a rate of $\lambda = 4$ times per hour, you expect to wait $\frac{1}{4}$ of an hour (15 minutes) for the next bus).
• Variance: $\text{Var}(X) = \frac{1}{\lambda^2}$

Real-World Examples

1. Waiting Times: The time you wait until the next customer calls the support desk.
2. Reliability Engineering: The amount of time until a mechanical part fails (assuming a constant failure rate).
3. Radioactive Decay: The time until a radioactive atom decays.

Interactive Visualization

Use the slider below to adjust the rate parameter ($\lambda$). Notice how the density is highly concentrated near zero, meaning shorter wait times are always more probable than longer wait times. As the rate $\lambda$ increases, the curve decays even faster.

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