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**The Comer Lectures on Random Variables and Signals**

Topic 8: Functions of Random Variables

We often do not work with the random variable we observe directly, but with some function of that random variable. So, instead of working with a random variable X, we might instead have some random variable Y=g(X) for some function g:**R** → **R**.

In this case, we might model Y directly to get f$ _Y $(y), especially if we do not know g. Or we might have a model for X and find f$ _Y $(y) (or p$ _Y $(y)) as a function of f$ _X $ (or p$ _X $ and g.

We will discuss the latter approach here.

More formally, let X be a random variable on (*S,F*,P) and consider a mapping g:**R** → **R**. Then let Y$ (\omega)= $g(X($ \omega)) $ ∀$ \omega $ ∈ *S*.

We normally write this as Y=g(X).

Graphically,

Is *Y* a random variable? We must have Y$ ^{-1} $(A) ≡ {$ \omega $ ∈ *S*: Y$ (\omega) $ ∈ A} = {$ \omega $ ∈ *S*: g(X$ (\omega) $) ∈ A} be an element of *F* ∀A ∈ B(**R**) (Y must be Borel measurable).

We will only consider functions g in this class for which Y$ ^{-1} $(A) ∈ *F* ∀A ∈ B(**R**), so that if Y=g(X) for some random variable X, Y will be a random variable.

What is the distribution of Y? Consider 3 cases:

- X discrete, Y discrete
- X continuous, Y discrete
- X continuous, Y continuous

Note: you cannot have a continuous Y from a discrete X.

## Contents

## Case 1: X and Y Discrete

Let $ R_X $ ≡ X(*S*) be the range space of X and $ R_Y $ ≡ g(X(*S*)) be the range space of Y (i.e. the image of X(*S*) under g). Then the pmf of Y is

But this means that

**Example** $ \quad $ Let X be the value rolled on a die and

Then *R*$ _X $ = {0,1,2,3,4,5,6} and *R*$ _Y $ = {0,1} and g(x) = x % 2.

## Case 2: X Continuous, Y Discrete

The pmf of Y in this case is

**R**: g(x)=y} ∀y ∈

*R*$ _y $

i.e. for a given y ∈ *R*$ _y $, D$ _y $ is the set of all x ∈ **R** such that g(x) = y.

Then,

**Example** Let g(x) = u(x - x$ _0 $) for some x$ _0 $ ∈ **R**, and let Y=g(X). Then $ R_Y $ = {0,1} and

**R**: x < x$ _0 $} = (-∞, x$ _0 $)

**R**: x ≥ x$ _0 $} = [ x$ _0 $, ∞)

So,

## Case 3: X and Y Continuous

We will discuss 2 methods for finding f$ _Y $ in this case.

**Approach 1**

First, find the cdf F$ _Y $.

where D$ _y $ = {x ∈

**R**: g(x) ≤ y}.

i.e. for a given y ∈ **R**, D$ _y $ is the set of all x ∈ **R** such that g(x) ≤ y.

Then

Differentiate F$ _Y $ to get f$ _y $.

You can find D$ _Y $ graphically or analytically

**Example**

For y = y$ _1 $ and y = y$ _2 $,

Then

**Example** Y = aX + b, a,b ∈ **R**, a ≠ 0

So,

Then

**Example** Y = X$ ^2 $

For y < 0, D$ _y $ = ø

For y ≥ 0,

So,

and

For general y, we need to find subsets of the y-axis that have solutions of the same form and solve the problems separately for the different subsets.

**Approach 2**

Use a formula for f$ _y $ in terms of f$ _X $. To derive the formula, assume the inverse function g$ ^{-1} $ exists, so if y = g(x), then x = g$ ^{-1} $(y). Also assume g and g$ ^{-1} $ are differentiable. Then, if Y = g(X), we have that

**Proof:**

First consider g monotone (strictly monotone) increasing (note that for differentiable and hence continuous functions defined for a given interval, injection implies monotonicity, hence it is sufficient to limit our analysis to monotonic functions only).

Since {y < Y ≤ y + Δy} = {x < X ≤ x + Δx}, we have that P(y < Y ≤ y + Δy) = P(x < X ≤ x + Δx).

Use the following approximations:

- P(y < Y ≤ y + Δy) ≈ f$ _Y $(y)Δy
- P(x < X ≤ x + Δx) ≈ f$ _X $(x)Δx

Since the left hand sides are equal,

Now as Δy → 0, we also have that Δx → 0 since g is continuous, and the approximations above become equalities. We rename Δy, Δx as dy and dx respectively, so letting Δy → 0, we get

We normally write this as

A similar derivation for g monotone decreasing gives us the general result for invertible g:

Note this result can be extended to the case where y = g(x) has n solutions x$ _1 $,...,x$ _n $, in which case,

For example, if Y = X$ ^2 $,

## References

- M. Comer. ECE 600. Class Lecture. Random Variables and Signals. Faculty of Electrical Engineering, Purdue University. Fall 2013.

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