The Difference of Two Independent Exponential Random Variables

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?