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Probability · Lecture 39 of 76 · 9:05

Recitation: The PDF of the Absolute Value of X

The Probability Distribution Function (PDF) of [X] on YouTube

Study guide

What this lecture covers

This short recitation problem shows how to find the PDF of Y = |X| given the PDF of X, using two concrete examples before generalizing to a formula. It builds directly on the derived-distributions method, applied to the specific and commonly useful case of an absolute value.

The video reasons first from the discrete case, where a value of |X| can come from two different values of X, then transfers that reasoning to continuous densities.

Key ideas

  • Two contributions per value: just as in the discrete case where P(|X|=v) combines the probability at X=v and X=-v, the density of Y=|X| at a point y combines the density of X at y and at -y.
  • Fold-and-stack visualization: graphically, taking the PDF of X, flipping the negative-side portion over the y-axis, and stacking it on top of the positive-side portion gives the PDF of |X|.
  • One-sided densities need no folding: if X is already always non-negative, then |X| = X and the PDF is unchanged.
  • General formula: the PDF of Y=|X| equals the PDF of X at y plus the PDF of X at -y, for y >= 0.

Walkthrough

A two-piece uniform example (0:01)

For X uniform with density 1/3 on [-2,1], the video reasons that the density of Y=|X| at a given value has two contributions, one from the positive and one from the negative side of X's range. Folding the negative portion onto the positive portion and stacking gives a two-level step function: density 2/3 for y between 0 and 1 (where both sides overlap), and density 1/3 for y between 1 and 2 (where only the negative side contributes).

A one-sided exponential example (4:03)

For X exponential with rate 2 (already non-negative), there is nothing on the negative side to fold, so |X| has exactly the same PDF as X.

The general formula (6:11)

Generalizing both examples, the PDF of Y=|X| at y is the PDF of X at y plus the PDF of X at -y, mirroring the discrete formula that adds P(X=y) and P(X=-y).

Before you watch

  • Review the general method for finding the distribution of a function of a random variable.
  • It helps to first think through the discrete analog of taking an absolute value before moving to densities.

Check your understanding

  1. Why does the density of |X| at a point y generally involve two terms rather than one?
  2. Why does the PDF of |X| equal the PDF of X unchanged when X is already non-negative?
  3. In the uniform example, why is the density of |X| higher between 0 and 1 than between 1 and 2?

Vocabulary

absolute value (phrase)
The distance of a number from zero, always positive or zero.
We find the PDF of the absolute value of X.
fold (verb)
To bend or flip one part of a shape over onto another part.
We fold the negative side of the density onto the positive side.
stack (verb)
To place one thing directly on top of another.
Stacking the two folded parts gives the new density.
one-sided (adjective)
Existing only on one side, with nothing on the other.
A one-sided density needs no folding since there is nothing negative.
step function (phrase)
A function that changes value suddenly at certain points, staying flat in between.
The resulting density is a two-level step function.
overlap (verb)
To share the same space or range as something else.
The two sides overlap between 0 and 1.
generalize (verb)
To form a broader rule from specific examples.
We generalize both examples into one formula.
derived distribution (phrase)
The probability distribution of a new variable built as a function of another random variable.
This problem is a derived-distribution problem, finding the PDF of |X|.
PDF (phrase)
A curve that describes how likely different values of a continuous variable are.
We are given the PDF of X and want the PDF of |X|.
density (noun)
The value of a PDF at a specific point, showing how likely values are near that point.
The density of Y at y combines two contributions from X.
contribution (noun)
A separate part that adds to a total result.
The density of |X| has two contributions, one from each sign of X.
symmetric (adjective)
Having matching shape on both sides.
A one-sided variable becomes trivially symmetric once folded.
visualization (noun)
A way of picturing an idea using a diagram or image.
The fold-and-stack visualization makes the formula easy to remember.
transfer (verb)
To move an idea or method from one setting to another.
The lecture transfers reasoning from the discrete case to densities.
graphically (adverb)
By using a picture or graph rather than only equations.
Graphically, we flip the negative side of the PDF onto the positive side.
discrete (adjective)
Made of separate, countable values rather than a smooth range.
The reasoning starts from the discrete case before moving to densities.
mirror (verb)
To closely match or reflect something else.
The continuous formula mirrors the discrete one.
combine (verb)
To join separate parts together into one result.
We combine the two densities at y and -y.
non-negative (adjective)
Never less than zero.
An exponential random variable is already non-negative.
concrete (adjective)
Specific and clear, not vague or general.
Two concrete examples are worked before the general formula.
transform (verb)
To change a variable into a new one using a function.
Taking the absolute value transforms X into Y.
method (noun)
A planned way of doing something.
This is a specific case of the general derived-distributions method.
reason (verb)
To think through a problem logically, step by step.
The video reasons first from the discrete case.
analog (noun)
A version of something in a different but related setting.
The discrete analog helps before moving to continuous densities.
commonly useful (phrase)
Often applied in practice because it comes up so frequently.
Finding the PDF of |X| is a commonly useful special case.

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
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