Competing Exponentials

What this lecture covers

This worked problem finds the probability that three independent exponential random variables satisfy X < Y < Z, where each variable has its own rate parameter. Rather than integrating directly, the lecture reinterprets each exponential as the first arrival time of a Poisson process, turning the ordering question into a question about which process "wins the race" to arrive first.

You should already understand exponential random variables and Poisson process merging before watching. Afterward, you'll be able to compute ordering probabilities among competing exponentials using process merging and the fresh-start property, and sanity-check the result against a symmetric special case.

Key ideas

• Exponential as first arrival: an exponential random variable with rate lambda can be viewed as the first arrival time of a Poisson process with that rate.
• Splitting the event: X < Y < Z is rewritten as X < min(Y,Z) and Y < Z, two events shown to be independent.
• Merged process rate: merging independent Poisson processes with rates lambda, mu, nu gives a process with rate lambda + mu + nu.
• Probability the merged winner is from a given process: the probability the first arrival of a merged process comes from process X is lambda / (lambda + mu + nu).
• Fresh-start property: after any arrival, the remaining Poisson processes restart independently, so what happens after the first arrival is independent of which process arrived first.
• Sanity check with symmetry: when all three rates are equal, the answer reduces to 1/6, matching the intuition that all six orderings of three symmetric processes are equally likely.

Before you watch

• Review exponential random variables and their memoryless property.
• Know how merging independent Poisson processes works and how the merged rate is formed.
• Be familiar with conditional independence arguments for splitting a joint event into a product of probabilities.

Check your understanding

1. Why can P(X < Y < Z) be split into P(X < min(Y,Z)) times P(Y < Z)?
2. What role does the fresh-start property play in showing the two split events are independent?
3. How would the formula change if you wanted P(Y < X < Z) instead?
4. Why does the symmetric case (lambda = mu = nu) give exactly 1/6?