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Probability · Lecture 45 of 76 · 11:30

The Variance in the Stick Breaking Problem

The Variance in the Stick Breaking Problem on YouTube

Study guide

What this lecture covers

This is a worked recitation problem applying two tools from the main lecture on iterated expectations: the law of iterated expectations and the law of total variance. It answers a concrete question: if a stick of length l is broken uniformly at random, and the remaining left piece is broken uniformly at random again, what are the expectation and variance of the final piece's length?

The problem shows how conditioning on the first break turns a hard two-stage calculation into two easy conditional calculations combined with the two laws. After watching, you should be able to identify what to condition on to simplify a nested random process, and apply both laws to compute a mean and a variance in stages.

Key ideas

  • Condition on the useful intermediate variable: letting Y be the length after the first break makes the second break simple to analyze, since X is uniform between 0 and Y once Y is known.
  • Conditional expectation and variance become simple formulas: given Y = y, E[X|Y=y] = y/2 and Var(X|Y=y) = y^2/12, the standard formulas for a variable uniform on [0, y].
  • Law of iterated expectations applied: E[X] = E[E[X|Y]] = E[Y/2] = E[Y]/2 = l/4, since Y itself is uniform on [0, l].
  • Law of total variance applied: Var(X) = E[Var(X|Y)] + Var(E[X|Y]), computed here as l^2/36 + l^2/48 = 7l^2/144.
  • Choosing the conditioning variable matters: not every choice of variable to condition on simplifies the problem; picking a useful one is a skill built through practice.

Before you watch

  • Watch the main lecture on iterated expectations first, since this problem directly applies the law of iterated expectations and the law of total variance introduced there.
  • Know the mean and variance formulas for a uniform random variable.

Check your understanding

  1. Why does conditioning on the length after the first break make the second stage of the problem easy to analyze?
  2. Derive E[X] = l/4 using the law of iterated expectations.
  3. Why does the law of total variance need two separate terms, and what does each term represent in this problem?
  4. What criteria make a variable a good choice to condition on when simplifying a multi-stage random process?

Vocabulary

intermediate variable (phrase)
A variable in the middle of a process, used to connect an earlier step to a later one.
Y is a useful intermediate variable between the first and second break.
nested (adjective)
Contained inside another similar structure, like a smaller box inside a bigger one.
This is a nested random process: one break happens inside the result of another.
law of iterated expectations (phrase)
The rule that averaging a conditional expectation over all cases gives the overall expectation.
We apply the law of iterated expectations to find E[X].
law of total variance (phrase)
A rule for splitting overall variance into two parts based on conditioning.
The law of total variance gives the final variance of the stick's length.
stages (noun)
Separate steps that happen one after another in a process.
The problem is solved in two easy stages instead of one hard one.
variance (noun)
A number that measures how spread out a random variable's values are.
We compute the variance of the final piece's length.
expectation (noun)
The average value a random variable would take if the experiment were repeated many times.
The expectation of the final piece is l/4.
uniform distribution (phrase)
A pattern where every value in a range is equally likely.
The stick is broken according to a uniform distribution.
conditioning (noun)
Assuming a value is known in order to make a calculation simpler.
Conditioning on the first break simplifies the second stage.
worked example (phrase)
A full example solved step by step to show how a method is used.
This lecture is a worked example applying two laws from probability.
recitation (noun)
A class session that practices ideas from the main lecture with example problems.
This recitation problem uses the stick-breaking setup.
derive (verb)
To work out a result step by step from known rules or facts.
We derive E[X] = l/4 using the law of iterated expectations.
formula (noun)
A fixed rule written with symbols that gives a result from given values.
The mean and variance formulas for a uniform variable are used twice here.
concrete (adjective)
Specific and clear, not vague or general.
The problem answers a concrete question about the final piece's length.
multi-stage (adjective)
Made of several separate steps done one after another.
This is a multi-stage random process with two breaks.
remaining (adjective)
Left over after part of something has been used or removed.
The remaining left piece is broken a second time.
tool (noun)
A method or technique used to solve a problem.
This problem applies two tools from the main lecture.
calculation (noun)
The process of working out a number using math.
Conditioning turns a hard calculation into two easy ones.
straightforward (adjective)
Simple and easy to do, without complications.
Once you condition on Y, the second stage becomes straightforward.
skill (noun)
An ability gained through practice.
Picking a useful variable to condition on is a skill built through practice.
combine (verb)
To join separate parts together into one result.
We combine the two conditional calculations using the two laws.
simplify (verb)
To make something easier to understand or solve.
Conditioning on the first break helps simplify a nested random process.
criteria (noun)
The standards used to judge or choose something.
What criteria make a variable a good choice to condition on?
identify (verb)
To recognize and name something correctly.
You should be able to identify what to condition on.
random variable (phrase)
A quantity whose value depends on the outcome of a random experiment.
X and Y are both random variables in this problem.

Chapters

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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