Chapter 5A: Continuous Random Variables, PDFs, and Uniform Distribution

MATH/STAT 394: Probability I

Arman Jahangiri

University of Washington Department of Mathematics

Summer 2026

Arman Jahangiri

Chapter 5A: Continuous Random Variables, PDFs, and Uniform Distri bution

Summer 2026

1

Learning Goals

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

▶ deﬁne and interpret PDFs and CDFs;

▶ compute probabilities by integrating a PDF;

▶ recognize valid PDFs;

▶ compute expectation for continuous random variables;

▶ deﬁne and use the Uniform distribution;

▶ understand the universality of the Uniform distribution.

Arman Jahangiri

Chapter 5A: Continuous Random Variables, PDFs, and Uniform Distri bution

Summer 2026

2

Continuous Random Variables

Arman Jahangiri

Chapter 5A: Continuous Random Variables, PDFs, and Uniform Distri bution

Summer 2026

3

From Discrete to Continuous Random Variables

Discrete random variables take values in a ﬁnite or countably inﬁnite set.

Continuous random variables can take any value in an interval, such as

(0, 1),

(a, b),

[0, ∞),

R.

Main diﬀerence

For a continuous random variable,

P(X = x) = 0

for every individual point x.

Arman Jahangiri

Chapter 5A: Continuous Random Variables, PDFs, and Uniform Distri bution

Summer 2026

4

Probability Density Functions

Deﬁnition

For a continuous random variable X, the probability density function, or PDF, is a function f such that

Z

P(X ∈ A) =

f (x) dx, ∀A ⊆ R

A

The value f (x) itself is not a probability (could be greater than 1).

Probabilities are areas under the density curve.

Arman Jahangiri

Chapter 5A: Continuous Random Variables, PDFs, and Uniform Distri bution

Summer 2026

5

Valid PDFs

A function f is a valid PDF if:

f (x) ≥ 0

for all x,

and

Z

∞

_−∞f (x) dx = 1.

Analogy

For discrete random variables:

X

p_X(x) ≥ 0,

p_X(x) = 1.

x

For continuous random variables:

Z

∞

f_X(x) ≥ 0,

_−∞f (x) dx = 1.

Arman Jahangiri

Chapter 5A: Continuous Random Variables, PDFs, and Uniform Distri bution

Summer 2026

6

CDF and PDF

CDF has the same formula for continuous and discrete r.v.s, i.e.

F(x) = P(X ≤ x).

For a continuous random variable with PDF f , according to the deﬁnition of PDF,

Z

x

F(x) = _−∞f (t) dt.

Note that, according to the deﬁnition of CDF,

P(a < X ≤ b) = F(b) − F(a)

This, together with the Fundamental Theorem of Calculus, implies that

d

F(x) = F^′(x) = f (x)

(1)

dx

For continuous r.v.s, since P(X = a) = 0 for any a ∈ R,

P(X ≤ a) = P(X < a),

P(X ≥ a) = P(X > a),

P(a ≤ X ≤ b) = P(a < X ≤ b) = P(a ≤ X < b) = P(a < X < b).

Arman Jahangiri

Chapter 5A: Continuous Random Variables, PDFs, and Uniform Distri bution

Summer 2026

7

What Does the Density f (x) Mean?

For a continuous random variable,

Z

x

F(x) = P(X ≤ x) = _−∞f (t) dt.

For a small number ϵ > 0,

P(x ≤ X ≤ x + ϵ) = F(x + ϵ) − F(x).

Therefore,

P(x ≤ X ≤ x + ϵ)

F(x + ϵ) − F(x)

=

.

ϵ

Taking the limit as ϵ → 0,

By the deﬁnition of derivative,

Thus,

P(x ≤ X ≤ x + ϵ)

F(x + ϵ) − F(x)

lim

ϵ→0

= lim

.

ϵ→0

ϵ

F(x + ϵ) − F(x)

f (x) = F^′(x) = lim

.

ϵ→0

ϵ

P(x ≤ X ≤ x + ϵ)

f (x) = lim

ϵ→0

ϵ

Arman Jahangiri

Chapter 5A: Continuous Random Variables, PDFs, and Uniform Distri bution

Summer 2026

8

Expectation for Continuous Random Variables

Deﬁnition

If X is continuous with PDF f , then

Z

∞

E(X) = _−∞xf (x) dx.

If the support is an interval, we integrate over that interval.

Expectation is still the balancing point of the distribution.

Arman Jahangiri

Chapter 5A: Continuous Random Variables, PDFs, and Uniform Distri bution

Summer 2026

9

Continuous LOTUS

LOTUS

If X has PDF f and g : R → R, then

Z

∞

E(g(X)) = _−∞g(x)f (x) dx.

Proof idea. For a small interval [x, x + ∆x],

P(x ≤ X ≤ x + ∆x) ≈ f (x)∆x.

On this interval, g(X) is approximately g(x). Hence,

g(X)P(x ≤ X ≤ x + ∆x) ≈ g(x)f (x)∆x.

Adding over many small intervals,

X

E(g(X)) ≈

g(x_i)f (x_i)∆x.

i

Taking the limit as ∆x → 0, the Riemann sum becomes

Z

∞

E(g(X)) = _−∞g(x)f (x) dx.

Arman Jahangiri

Chapter 5A: Continuous Random Variables, PDFs, and Uniform Distri bution

Summer 2026 10

Uniform Distribution

Arman Jahangiri

Chapter 5A: Continuous Random Variables, PDFs, and Uniform Distri bution

Summer 2026 11

Uniform Distribution

Deﬁnition

A random variable U has the Uniform distribution on (a, b) if its PDF is



1



,

a < x < b,

b − a

f (x) =



0,

otherwise.

We write

U ∼ Unif(a, b).

PDF Veriﬁcation

1

f (x) =

> 0

b − a

since b > a. And

Z

∞

b

1

_−∞f (x) dx =

dx =

(b − a) = 1.

b − a

a

Arman Jahangiri

Chapter 5A: Continuous Random Variables, PDFs, and Uniform Distri bution

Summer 2026 12

Uniform CDF

If

U ∼ Unif(a, b),

then its CDF is





0,

x ≤ a,





x − a

b − a

1,

F(x) =

,

a < x < b,





x ≥ b.

For the standard Uniform,

we have

U ∼ Unif(0, 1),

F(x) = x, 0 < x < 1.

Arman Jahangiri

Chapter 5A: Continuous Random Variables, PDFs, and Uniform Distri bution

Summer 2026 13

Universality of the Uniform

Arman Jahangiri

Chapter 5A: Continuous Random Variables, PDFs, and Uniform Distri bution

Summer 2026 14

Why the Uniform Distribution Is Special

The Uniform distribution is a universal starting point for simulation.

Many algorithms ﬁrst generate

U ∼ Unif(0, 1),

then transform U to obtain another distribution.

Main idea

A Uniform random variable can be transformed into a random variable with many other distributions.

Arman Jahangiri

Chapter 5A: Continuous Random Variables, PDFs, and Uniform Distri bution

Summer 2026 15

Inverse CDF Method

Suppose F is a continuous, strictly increasing CDF.

Let

U ∼ Unif(0, 1).

Deﬁne

Then

X = F⁻¹(U).

X has CDF F.

This is called the inverse CDF method or inverse transform sampling.

Arman Jahangiri

Chapter 5A: Continuous Random Variables, PDFs, and Uniform Distri bution

Summer 2026 16

Why the Inverse CDF Method Works

Let

X = F⁻¹(U).

Then

P(X ≤ x) = P(F⁻¹(U) ≤ x).

Since F is increasing,

Therefore,

F

⁻¹(U) ≤ x

⇐⇒ U ≤ F(x).

P(X ≤ x) = P(U ≤ F(x)).

Because U ∼ Unif(0, 1),

Thus X has CDF F.

P(U ≤ F(x)) = F(x).

Arman Jahangiri

Chapter 5A: Continuous Random Variables, PDFs, and Uniform Distri bution

Summer 2026 17

Probability Integral Transform

Conversely, if X has continuous strictly increasing CDF F, then

F(X) ∼ Unif(0, 1).

Interpretation

Applying the CDF to a random variable converts its value into a percentile.

For example, if exam scores have CDF F, then F(X) is the percentile of a randomly selected student.

Arman Jahangiri

Chapter 5A: Continuous Random Variables, PDFs, and Uniform Distri bution

Summer 2026 18

Uniform Mean and Variance

If U ∼ Unif(a, b), then

1

f_U(u) =

,

a ≤ u ≤ b.

b − a

Mean

ꢀ

ꢁ

Z

b

1

u²

b²− a²

a + b

E(U) =

u

du =

=

.

b − a

2

2(b − a)

2

a

Variance

First,

Z

b

1

b³− a³

a²+ ab + b²

E(U²) =

u²

du =

=

.

b − a

3(b − a)

3

a

Thus,

ꢂ

ꢃ

2

a²+ ab + b²

a + b

(b − a)²

Var(U) = E(U²) − [E(U)]²=

−

=

.

3

2

12

Arman Jahangiri

Chapter 5A: Continuous Random Variables, PDFs, and Uniform Distri bution

Summer 2026 19

Location-Scale Transformation

If

U ∼ Unif(0, 1),

then

a + (b − a)U ∼ Unif(a, b).

This transforms:

(0, 1) −→ (a, b).

Useful idea

Start with the standard Uniform and shift/scale to get any Uniform interval.

Arman Jahangiri

Chapter 5A: Continuous Random Variables, PDFs, and Uniform Distri bution

Summer 2026 20

Summary

▶ Continuous random variables have probabilities over intervals, not points.

▶ A PDF integrates to probabilities:

Z

b

P(a < X < b) =

f (x) dx.

a

▶ A valid PDF is nonnegative and integrates to 1.

▶ The CDF is accumulated area:

Z

x

F(x) = _−∞f (t) dt.

▶ Uniform probabilities are proportional to interval length.

▶ The Uniform distribution is universal through the inverse CDF method.

Next lecture

Normal and Exponential distributions.

Arman Jahangiri

Chapter 5A: Continuous Random Variables, PDFs, and Uniform Distri bution

Summer 2026 21