Chapter 8A: Transformations and Convolutions

MATH/STAT 394: Probability I

Arman Jahangiri

University of Washington Department of Mathematics

Summer 2026

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026

1

Learning Goals

By the end of this lecture, students should be able to:

▶ compute distributions of transformed random variables;

▶ apply the change-of-variables formula;

▶ distinguish between discrete and continuous transformations;

▶ derive distributions using the CDF method;

▶ deﬁne convolutions;

▶ compute distributions of sums of independent random variables;

▶ derive Gamma and Triangle distributions from convolutions.

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026

2

Transformations of Random Variables

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026

3

Why Transform Random Variables?

Transformations appear everywhere in probability and statistics.

Examples:

▶ unit conversion;

▶ standardization;

▶ logarithms;

▶ sums and averages;

▶ squares and absolute values;

▶ maxima and minima.

Main question

If

Y = g(X),

what is the distribution of Y ?

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026

4

Transformations in the Discrete Case

If X is discrete and

Y = g(X),

then

X

P(Y = y) =

P(X = x).

x:g(x)=y

One-to-one case

If g is one-to-one, then

P(Y = y) = P(X = g⁻¹(y)).

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026

5

Transformation in the Continuous Case

Setup

Let X be a continuous random variable with PDF f_Xand support

S_X= {x : f_X(x) > 0}.

Let

Y = g(X),

where g : S_X→ S_Yis diﬀerentiable and strictly monotone (i.e., strictly increasing or strictly decreasing). Then g has

an inverse

g⁻¹: S_Y→ S_X.

We write

x = g⁻¹(y).

The support of Y is

S_Y= g(S_X) = {g(x) : x ∈ S_X}.

In the next slide, we will see how to ﬁnd f_Y

.

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026

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Change of Variable Formula

Change of Variables Formula

For y ∈ S_Y,

ꢂ

ꢀ

ꢁ

d

f_Y(y) = f_X

g

⁻¹(y)

g⁻¹(y)

dy

and for y ∈/ S_Y,

f_Y(y) = 0.

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026

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Proof: g Increasing

Assume g is strictly increasing on S_X. Then

Y = g(X)

S_Y= g(S_X),

has support

and for y ∈ S_Y

,

Y ≤ y ⇐⇒ g(X) ≤ y ⇐⇒ X ≤ g⁻¹(y).

Therefore, since g is increasing,

ꢀ

ꢁ

ꢀ

ꢁ

F_Y(y) = P(Y ≤ y) = P(g(X) ≤ y) = P X ≤ g⁻¹(y) = F_Xg⁻¹(y) .

Now diﬀerentiate with respect to y. According to the chain rule, we have

ꢀ

ꢁ

d

f_Y(y) =

F_Y(y) =

F_Xg⁻¹(y) .

dy

Using the chain rule,

ꢀ

ꢁ

d

f_Y(y) = f_Xg⁻¹(y)

g⁻¹(y).

dy

Since g is increasing,

d

g⁻¹(y) > 0,

dy

so

ꢂ

ꢀ

ꢁ

d

f_Y(y) = f_Xg⁻¹(y)

g⁻¹(y) .

ꢂ

dy

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026

8

Proof: g Decreasing

Assume g is strictly decreasing on S_X. Then for y ∈ S_Y

,

Y ≤ y ⇐⇒ g(X) ≤ y.

Because g is decreasing, applying g⁻¹reverses the inequality:

g(X) ≤ y ⇐⇒ X ≥ g⁻¹(y).

Therefore,

ꢀ

ꢁ

ꢀ

ꢁ

F_Y(y) = P(Y ≤ y) = P X ≥ g⁻¹(y) = 1 − F_Xg⁻¹(y) .

Diﬀerentiate with respect to y,

ꢃ

ꢀ

ꢁ

ꢁꢄ

d

f_Y(y) =

1 − F_Xg⁻¹(y)

.

dy

Using the chain rule,

Since g is decreasing,

Hence,

d

f_Y(y) = −f_Xg⁻¹(y)

g⁻¹(y).

dy

d

g⁻¹(y) < 0.

dy

ꢂ

d

−

g⁻¹(y) =

g⁻¹(y) .

ꢂ

dy

Thus,

ꢂ

ꢀ

ꢁ

d

f_Y(y) = f_Xg⁻¹(y)

g⁻¹(y) .

ꢂ

dy

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026

9

Transformation Formula: Final Form

Let

Y = g(X),

where g : S_X→ S_Yis diﬀerentiable and strictly monotone (i.e., strictly increasing or strictly decreasing).

Then for y ∈ S_Y

,

ꢂ

dx

dy

f_Y(y) = f_X(x)

where

x = g⁻¹(y).

Equivalently,

ꢂ

ꢀ

ꢁ

d

f_Y(y) = f_Xg⁻¹(y)

g⁻¹(y) .

ꢂ

dy

For y ∈/ S_Y

,

f_Y(y) = 0.

Important

The absolute value appears because the inverse derivative is positive in the increasing case and negative in the

decreasing case.

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026 10

How to Use the Formula

Steps:

1

Write

y = g(x).

2

Solve for

x = g⁻¹(y).

3

Compute

ꢂ

dx

dy

.

4

Substitute everything in terms of y.

Do not forget

Always specify the support of the transformed variable.

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026 11

Log-Normal Distribution

Let

X ∼ N(0, 1),

and deﬁne

Y = e^X.

Here, g(x) = e^xis strictly increasing, so it satisﬁes the condition of the Change of Variable Formula. Note that

x = log y.

This means that

g

⁻¹(y) = log(y)

Also,

dx

dy

1

=

.

y

Thus,

1

f_Y(y) = φ(log y)

,

y > 0.

y

This is called the Log-Normal distribution.

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026 12

Squaring a Standard Normal

Let

X ∼ N(0, 1),

Y = X².

and deﬁne

The function

is not one-to-one.

g(x) = x²

So the simple change-of-variables formula does not directly apply.

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026 13

CDF Method for Y = X²

For

y > 0,

the event

is equivalent to

Thus,

X²≤ y

√

−

y ≤ X ≤ y.

√

F_Y(y) = Φ( y) − Φ(− y).

Using symmetry,

√

F_Y(y) = 2Φ( y) − 1.

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026 14

PDF of Y = X²

Diﬀerentiating,

Hence,

√

1

f_Y(y) = 2φ( y) ·

.

√

2

y

√

φ( y)

f_Y(y) =

,

y > 0.

√

y

This is an example of a Chi-Square distribution.

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026 15

Location-Scale Transformations

Suppose

Then

Y = a + bX,

y − a

b

̸= 0.

x =

,

b

and

ꢂ

dx

dy

1

=

.

|b|

Thus,

ꢃ

ꢄ

y − a

1

f_Y(y) = f_X

.

b

|b|

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026 16

Convolutions

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026 17

What Is a Convolution?

A convolution is the distribution of a sum of independent random variables.

If

T = X + Y ,

and X, Y are independent, we want the distribution of T.

Why sums matter

Sums appear everywhere:

▶ totals;

▶ averages;

▶ waiting times;

▶ counting processes.

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026 18

Discrete Convolution Formula

Theorem

If X and Y are independent discrete random variables, then

X

P(T = t) =

P(Y = t − x)P(X = x).

x

This is just the law of total probability.

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026 19

Continuous Convolution Formula

Theorem

If X and Y are independent continuous random variables, then

Z

∞

f_T(t) =

f_Y(t − x)f_X(x) dx.

−∞

This is the continuous analog of the discrete convolution sum.

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026 20

Convolution of Two Exponentials

Let

i.i.d.

X, Y ∼ Exp(λ).

Deﬁne

Then for

T = X + Y .

t > 0,

Z

t

f_T(t) =

λe^−λ(t−x)λe^−λxdx.

0

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026 21

Gamma Distribution Appears

Simplifying,

Therefore,

Z

t

f_T(t) = λ²e^−λt

dx.

0

f_T(t) = λ²te^−λt

,

t > 0.

This is a Gamma distribution.

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026 22

Convolution of Two Uniforms

Let

i.i.d.

X, Y ∼ Unif(0, 1).

Deﬁne

Then

T = X + Y .

0 < T < 2.

The PDF becomes piecewise linear.

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026 23

Triangle Distribution

The PDF of

is

T = X + Y



t,

0 < t ≤ 1,





f_T(t) =

2 − t, 1 < t < 2,





0,

otherwise.

Interpretation

Middle values are more likely because there are more ways to obtain them.

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026 24

Why the Triangle Shape Makes Sense

Values near

1

can occur in many ways:

0.4 + 0.6,

0.2 + 0.8,

0.5 + 0.5,

etc.

But values near

0

2

or

require both variables to be extreme simultaneously.

This explains the triangular shape.

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026 25

Connection with Dice

Recall: the sum of two fair dice also produced a triangular PMF.

Analogy

Discrete Uniform −→ triangular PMF,

Continuous Uniform −→ triangular PDF.

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026 26

Summary

▶ Transformations convert one distribution into another.

▶ Continuous transformations require Jacobians.

▶ The CDF method always works.

▶ Convolutions describe sums of independent random variables.

▶ Sums of Exponentials lead to Gamma distributions.

▶ Sums of Uniforms lead to Triangle distributions.

Next lecture

Beta and Gamma distributions.

Arman Jahangiri

Chapter 8A: Transformations and Convolutions

Summer 2026 27