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Probability · Lecture 42 of 76 · 6:12

The Difference of Two Independent Exponential Random Variables

The Difference of Two Independent Exponential Random Variables on YouTube

Study guide

What this lecture covers

This is a worked recitation problem from MIT 6.041SC's derived distributions unit. It answers a specific question: if Romeo and Juliet each arrive at independent, identically distributed exponential times, what is the distribution of the difference between their arrival times? The problem applies the convolution formula from the main lecture on derived distributions to a subtraction instead of a sum.

After watching, you should be able to adapt the convolution formula to a difference of two random variables, split the calculation into cases based on the sign of the result, and use a symmetry argument to avoid solving the integral twice.

Key ideas

  • Rewriting a difference as a sum: Z = X - Y can be treated as X + (-Y), so the convolution formula for sums applies once f_Y is replaced with the density of -Y.
  • Density of a negated variable: f_{-Y}(z - x) = f_Y(x - z), since -Y = -1 occurring is the same event as Y = 1 occurring.
  • Case split by sign: the integral is worked out directly for z < 0; the case z >= 0 is then obtained from symmetry rather than a second integration.
  • Symmetry from i.i.d. variables: because X and Y are independent and identically distributed, X - Y and Y - X have the same distribution, so f_Z(z) = f_Z(-z).
  • Result: the difference of two independent exponential random variables with parameter lambda has density f_Z(z) = (lambda/2) * e^(-lambda * |z|), known as a Laplace (double exponential) distribution.

Before you watch

  • Know the convolution formula for the sum of two independent random variables, and the density of an exponential random variable.
  • Watching the main lecture on derived distributions (covering convolution) first makes this problem easier to follow.

Check your understanding

  1. Why can the formula for a sum of two random variables be reused for a difference by substituting -Y for Y?
  2. Work through why f_{-Y}(z - x) equals f_Y(x - z).
  3. Why does the i.i.d. assumption on X and Y let you skip directly computing the density for z >= 0?
  4. What shape does the resulting density have, and why is it symmetric around zero?

Vocabulary

i.i.d. (independent and identically distributed) (phrase)
Random variables that are independent of each other and follow the exact same distribution.
Romeo and Juliet's arrival times are i.i.d. exponential.
negate (verb)
To change the sign of a number, from positive to negative or the opposite.
We negate Y to rewrite the difference as a sum.
case split (phrase)
Dividing a problem into separate situations based on a condition.
The solution uses a case split based on the sign of z.
sign (noun)
Whether a number is positive or negative.
The case split depends on the sign of the result.
Laplace distribution (phrase)
A distribution shaped like two mirrored exponential curves meeting at a point, also called a double exponential.
The difference of two exponentials follows a Laplace distribution.
double exponential (phrase)
Another name for a distribution formed from two mirrored exponential curves.
The double exponential density is symmetric around zero.
convolution (noun)
A formula for combining two densities to find the distribution of their sum.
The convolution formula for sums is adapted here for a difference.
density (noun)
The value of a PDF at a specific point, showing how likely values are near that point.
The final density is (lambda/2) * e^(-lambda*|z|).
symmetry argument (phrase)
A reasoning method that uses matching structure to avoid repeating a calculation.
A symmetry argument gives the z >= 0 case for free.
adapt (verb)
To change a method slightly so it fits a new situation.
We adapt the convolution formula to a difference of two variables.
substitute (verb)
To replace one thing with another in a formula.
We substitute -Y in place of Y in the convolution formula.
rewrite (verb)
To express something again in a different but equivalent form.
We rewrite the difference Z = X - Y as X + (-Y).
integration (noun)
The mathematical process of adding up a continuous quantity.
Symmetry avoids a second round of integration.
avoid (verb)
To stay away from doing something unnecessary.
The symmetry argument helps avoid solving the integral twice.
mirrored (adjective)
Reflected, matching another shape as if seen in a mirror.
The Laplace distribution is built from two mirrored exponential curves.
formula (noun)
A fixed rule written with symbols that gives a result from given values.
The convolution formula is reused here for a subtraction.
derive (verb)
To work out a result step by step from known rules or facts.
We derive the density of the difference between two arrival times.
reuse (verb)
To use something again for a new purpose.
We reuse the sum formula by treating a difference as a sum of a negated variable.
closed-form (adjective)
Expressed with a single explicit formula rather than left unsolved.
The problem ends with a closed-form density for Z.
exponential random variable (phrase)
A continuous random variable often used to model waiting times, with a constant rate.
Romeo and Juliet's arrival times are each an exponential random variable.
arrival time (phrase)
The moment when an event, such as a person showing up, happens.
The problem compares the two arrival times of Romeo and Juliet.
recitation problem (phrase)
An example problem worked in a practice class to apply lecture ideas.
This is a worked recitation problem from the derived distributions unit.
event (noun)
A specific outcome or set of outcomes being considered.
The event z < 0 needs its own part of the calculation.
parameter (noun)
A fixed number that controls the shape of a distribution.
Both exponentials share the same rate parameter, lambda.
shape (noun)
The overall visual form a curve or distribution has.
The resulting density has the shape of two mirrored exponential curves.

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: Kuang Xu

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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