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Probability · Lecture 53 of 76 · 7:42

Competing Exponentials

Study guide

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

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

Vocabulary

compete (verb): To try to win against others in a race or contest.
Three exponentials compete to be the smallest.
race (noun): A contest to see which of several things finishes first.
Each process is like a racer trying to arrive first.
win (verb): To finish first or succeed over others.
The process with the smallest value wins the race.
reinterpret (verb): To understand or explain something in a new way.
We reinterpret each exponential as a Poisson arrival time.
fresh-start property (phrase): The idea that after an event, a process behaves like it just began, with no memory of the past.
The fresh-start property lets us treat what happens after the first arrival independently.
reduce to (phrasal verb): To simplify down to a simpler case or number.
The symmetric case reduces to exactly 1/6.
ordering (noun): The sequence in which things happen or are arranged.
We want the probability of a specific ordering of X, Y, and Z.
exponential random variable (phrase): A continuous random variable often used to model waiting times, with a constant rate.
X, Y, and Z are each an independent exponential random variable.
rate parameter (phrase): The number that controls how fast an exponential random variable tends to occur.
Each exponential has its own rate parameter, lambda, mu, or nu.
independent (adjective): Not affected by, or related to, another variable.
The three exponential random variables are independent.
sanity check (phrase): A simple test to see whether a result makes sense.
The symmetric case works as a sanity check on the formula.
symmetric (adjective): Having the same structure or value on all sides.
The symmetric case has all three rates equal.
special case (phrase): A specific, simpler version of a more general problem.
Equal rates form a special case that is easy to check.
intuition (noun): A natural feeling for how something works, before working out the exact math.
The 1/6 answer matches the intuition that all orderings are equally likely.
joint event (phrase): An event that depends on two or more variables happening together.
X < Y < Z is a joint event involving three variables.
conditional independence (phrase): Independence between two events that holds once something else is known.
Conditional independence lets us split the joint event into two factors.
arrival time (phrase): The moment when the first event of a process happens.
Each exponential is viewed as the arrival time of a Poisson process.
minimum (noun): The smallest value among a group of numbers or variables.
X < min(Y,Z) is one of the two split events.
restart (verb): To begin again from a fresh state.
The remaining processes restart independently after the first arrival.
formula (noun): A fixed rule written with symbols that gives a result from given values.
The formula gives the probability that X wins the race.
integrate (verb): To add up a continuous quantity over a range of values.
The lecture avoids having to integrate directly over three variables.
product (noun): The result of multiplying two or more numbers together.
The joint probability is written as a product of two simpler probabilities.
memoryless (adjective): Having no dependence on how much time has already passed.
The exponential random variables are memoryless.
derive (verb): To work out a result step by step from known rules or facts.
We derive the probability that X is the smallest of the three.
split (verb): To divide something into separate, simpler parts.
We split the event X < Y < Z into two independent pieces.

From the YouTube description

MIT 6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013
View the complete course: http://ocw.mit.edu/6-041SCF13
Instructor: Jimmy Li

License: Creative Commons BY-NC-SA
More information at http://ocw.mit.edu/terms
More courses at http://ocw.mit.edu

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