The Sum of Discrete and Continuous Random Variables

What this lecture covers

This is a short worked problem from MIT 6.041SC's derived distributions unit. It answers a specific question left open by the main convolution lecture: what if you add a discrete random variable X and a continuous random variable Y, rather than two variables of the same type? The problem walks through the standard CDF-then-derivative method, conditioning on the discrete variable to make the calculation tractable.

After watching, you should be able to derive the density of a sum of a discrete and a continuous independent random variable, and recognize how the result relates to the convolution formula covered in the main lecture.

Key ideas

• Standard approach: find the CDF of Z = X + Y first, then differentiate to get the density.
• Condition on the discrete variable: applying the total probability theorem by conditioning on X (rather than the continuous Y) avoids the technical difficulties of conditioning on a continuous variable.
• Independence lets you drop the conditioning: once X is fixed, independence of X and Y means the conditional CDF of Y given X equals the plain CDF of Y.
• Differentiating a sum: after summing the conditional CDF terms over all values of x, term-by-term differentiation (justified formally for a finite, and informally for a countable, number of terms) yields the density of Z.
• Result resembles convolution: the final formula, f_Z(z) = sum over x of f_Y(z - x) * P_X(x), mirrors the convolution formula, with a PMF replacing one of the two densities that would appear in the fully continuous case.

Before you watch

• Know the CDF-then-derivative method for finding a density, and the convolution formula for sums of two independent continuous or two independent discrete random variables.
• Be comfortable applying the total probability theorem by conditioning on a random variable.

Check your understanding

1. Why does the problem condition on the discrete variable X rather than the continuous variable Y?
2. Why is it valid to drop the conditioning on X when writing the CDF of Y given X = x?
3. What role does independence of X and Y play in this derivation?
4. How does the resulting formula compare to the convolution formula for two continuous random variables?