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Probability · Lecture 43 of 76 · 5:37

The Sum of Discrete and Continuous Random Variables

Study guide

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

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

Vocabulary

tractable (adjective): Easy enough to handle or solve with the tools you have.
Conditioning on the discrete variable makes the problem tractable.
condition on (phrase): To assume a certain variable's value is known and calculate from there.
We condition on the discrete variable X first.
total probability theorem (phrase): A rule for finding an overall probability by adding contributions from each possible case.
The total probability theorem breaks the CDF into pieces over each value of X.
term-by-term (phrase): Handling each part of a sum one at a time, separately.
We differentiate the sum term-by-term to get the density.
countable (adjective): Able to be listed one by one, even if there are infinitely many.
X takes a countable number of possible values.
resemble (verb): To look similar to something else.
The final formula resembles the convolution formula.
convolution (noun): A formula for combining two distributions to find the distribution of their sum.
The result mirrors the convolution formula from the main lecture.
differentiate (verb): To take the derivative of a function.
We differentiate the CDF to get the density.
derivative (noun): A measure of how fast a function changes at a point.
Taking the derivative of the CDF gives the density function.
sum (noun): The result of adding two or more quantities together.
Z is defined as the sum of a discrete and a continuous variable.
formula (noun): A fixed rule expressed with symbols for calculating something.
The final formula gives the density of Z directly.
technical difficulty (phrase): A tricky or awkward detail that complicates a calculation.
Conditioning on a discrete variable avoids the technical difficulties of conditioning on a continuous one.
finite (adjective): Having a limited, countable number of parts, not endless.
Term-by-term differentiation is formally justified for a finite number of terms.
informally (adverb): In a loose way, without a fully strict proof.
The result is justified informally for a countable number of terms.
formally (adverb): In a strict, rigorous, mathematically proven way.
The step is formally justified only for a finite sum.
mirror (verb): To closely match or reflect the structure of something else.
The new formula mirrors the standard convolution formula.
fix (verb): To hold a variable's value constant for the rest of a calculation.
Once X is fixed, independence simplifies the conditional CDF.
plain (adjective): Simple and ordinary, without extra conditions.
The conditional CDF of Y given X equals the plain CDF of Y.
specific (adjective): Particular and clearly defined, not general.
This problem answers a specific question left open by the main lecture.
standard (adjective): Usual and widely accepted as the normal way of doing something.
The CDF-then-derivative method is the standard approach here.
worked problem (phrase): An example problem solved step by step to show the method.
This is a short worked problem from the derived distributions unit.
unit (noun): A section of a course covering one connected topic.
This problem comes from the course's derived distributions unit.
recognize (verb): To notice and identify something as familiar.
You should recognize how the result relates to convolution.
apply (verb): To use a method or rule in a specific situation.
The total probability theorem is applied by conditioning on X.
drop (verb): To remove something that is no longer needed.
Independence lets you drop the conditioning on X.
independent (adjective): Not affected by or related to another variable.
X and Y are assumed to be independent random variables.

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: Jagdish Ramakrishnan

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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