Chapter 8B: Beta and Gamma Distributions

MATH/STAT 394: Probability I

Arman Jahangiri

University of Washington Department of Mathematics

Summer 2026

Arman Jahangiri

Chapter 8B: Beta and Gamma Distributions

Summer 2026

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Learning Goals

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

▶ deﬁne the Beta distribution;

▶ understand how Beta parameters aﬀect shape;

▶ recognize Uniform distributions as special Beta distributions;

▶ deﬁne the Gamma function and Gamma distribution;

▶ interpret Gamma distributions as sums of Exponential random variables;

▶ compute means and variances of Beta and Gamma random variables;

▶ understand the relationship between Gamma and Poisson processes.

Arman Jahangiri

Chapter 8B: Beta and Gamma Distributions

Summer 2026

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Beta Distribution

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Why the Beta Distribution?

Many random quantities naturally live between

0

and 1.

Examples:

▶ probabilities;

▶ proportions;

▶ percentages;

▶ fractions;

▶ success rates.

Main idea

The Beta distribution is one of the most important distributions on the interval

(0, 1).

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Beta Function

Deﬁnition

For

a, b > 0,

the Beta function is

Z

1

B(a, b) =

x^a−1(1 − x)^b−1dx.

0

The Beta function serves as a normalization constant.

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Chapter 8B: Beta and Gamma Distributions

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Deﬁnition of the Beta Distribution

Deﬁnition

A random variable X has the Beta distribution with parameters

a, b > 0

if its PDF is

x^a−1(1 − x)^b−1

f (x) =

,

0 < x < 1.

B(a, b)

Notation:

X ∼ Beta(a, b).

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Shape of Beta Distributions

Diﬀerent parameter values create very diﬀerent shapes.

Examples:

▶

a = b = 1

gives a Uniform distribution;

▶

a, b > 1

produces mound-shaped densities;

▶

a < 1

or

b < 1

can create spikes near the endpoints.

Important

The Beta family is extremely ﬂexible.

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Uniform Distribution as a Beta Distribution

If

a = b = 1,

then

B(1, 1) = 1,

and the PDF becomes

Thus:

f (x) = 1,

0 < x < 1.

Beta(1, 1) = Unif(0, 1).

Arman Jahangiri

Chapter 8B: Beta and Gamma Distributions

Summer 2026

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Mean and Variance of Beta

If

X ∼ Beta(a, b),

then

a

E(X) =

.

a + b

Also,

ab

Var(X) =

.

(a + b)²(a + b + 1)

Interpretation

The parameters control both:

▶ the center;

▶ the concentration.

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Chapter 8B: Beta and Gamma Distributions

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Beta Distribution and Beliefs

Suppose a coin has unknown success probability

p.

Before observing data, we may model uncertainty about p using a Beta distribution.

Example:

p ∼ Beta(2, 2)

represents a prior belief favoring values near

0.5.

Bayesian interpretation

Beta distributions are natural models for uncertain probabilities.

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Chapter 8B: Beta and Gamma Distributions

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Beta-Binomial Connection

Suppose:

▶

p ∼ Beta(a, b),

and conditional on p,

▶

X | p ∼ Bin(n, p).

Then the posterior distribution of p given X = k is again Beta:

p | X = k ∼ Beta(a + k, b + n − k).

This is called conjugacy.

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Chapter 8B: Beta and Gamma Distributions

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Bayes’ Billiards

Imagine tossing a ball uniformly onto a billiard table.

A second sequence of balls lands independently.

We ask: what fraction land to the left of the ﬁrst ball?

This naturally leads to Beta distributions.

Historical note

This is one of the earliest Bayesian probability models.

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Gamma Function

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Why the Gamma Function?

Recall:

n! = 1 · 2 · 3 · · · n.

Can factorials be extended to non-integer values?

Answer

Yes. The Gamma function generalizes factorials to positive real numbers.

Deﬁnition

For

α > 0,

the Gamma function is

Z

∞

Γ(α) =

x^{α−1 −x}dx.

e

0

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Chapter 8B: Beta and Gamma Distributions

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Connection with Factorials

Theorem

For positive integers n,

Γ(n) = (n − 1)!.

Proof. Show Γ(n) = (n − 1) × Γ(n − 1) using integration by parts, and then solve the recursion with induction.

Examples:

Γ(1) = 1,

Γ(2) = 1!,

Γ(3) = 2!,

etc.

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Gamma Distribution

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Deﬁnition of Gamma Distribution

Deﬁnition

A random variable X has the Gamma distribution with parameters

α, λ > 0

if its PDF is

λ^α

f (x) =

x^α−1e^−λx

,

x > 0.

Γ(α)

Notation:

X ∼ Gamma(α, λ).

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Chapter 8B: Beta and Gamma Distributions

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Exponential Distribution as a Gamma Distribution

If

α = 1,

Γ(1) = 1,

then

and the PDF becomes

Thus:

f (x) = λe^−λx

.

Gamma(1, λ) = Exp(λ).

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Chapter 8B: Beta and Gamma Distributions

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Mean and Variance of Gamma

If

X ∼ Gamma(α, λ),

then

α

E(X) =

.

λ

Also,

α

λ²

Var(X) =

.

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Chapter 8B: Beta and Gamma Distributions

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Proof: Mean and Variance of Gamma

The density is

λ^α

f_X(x) =

x^α−1e^−λx

,

x > 0.

Γ(α)

Then

Z

∞

λ^α

E(X) =

xf_X(x) dx =

x^αe^−λxdx.

Γ(α)

0

Using

Z

∞

Γ(α + 1)

λ^α+1

x^αe^−λxdx =

,

0

we get

λ^α

Γ(α + 1)

λ^α+1

E(X) =

·

.

Γ(α)

Since Γ(α + 1) = αΓ(α),

α

.

E(X) =

λ

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Chapter 8B: Beta and Gamma Distributions

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Proof: Variance of Gamma

First compute

Z

∞

E(X²) =

x²f_X(x) dx.

0

Thus

Z

∞

λ^α

E(X²) =

x^{α+1 −λx}dx.

e

Γ(α)

0

Using

we get

Z

∞

Γ(α + 2)

λ^α+2

x^α+1

e

^−λxdx =

,

0

λ^α

Γ(α + 2)

λ^α+2

E(X²) =

·

.

Γ(α)

Since

Γ(α + 2) = (α + 1)αΓ(α),

we have

α(α + 1)

E(X²) =

.

λ²

Therefore,

α(α + 1)

α²

λ²

Var(X) = E(X²) − (E(X))²=

−

.

λ²

Hence

Arman Jahangiri

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^{Chapter 8}

V

^B:

a

^B

r

^et

(

^a

X

^an

)

^{d G}

=

^{amma Distributions}

α

Gamma as a Sum of Exponentials

Suppose

X₁, . . . , X_n

are independent Exponential(λ) random variables.

Deﬁne

T = X₁+ · · · + X_n.

Then:

T ∼ Gamma(n, λ).

Interpretation

Gamma distributions describe waiting times for multiple arrivals.

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Chapter 8B: Beta and Gamma Distributions

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Gamma and Negative Binomial

There is a strong analogy:

Discrete

Continuous

Interpretation

Geometric

Exponential

Gamma

waiting for ﬁrst event

Negative Binomial

waiting for multiple events

Big picture

Gamma distributions are continuous analogs of Negative Binomial distributions.

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Chapter 8B: Beta and Gamma Distributions

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MGF of Gamma Distribution

If

X ∼ Gamma(α, λ),

then its MGF is

ꢀ

ꢁ

α

λ

M_X(t) =

,

λ − t

for t < λ.

Arman Jahangiri

Chapter 8B: Beta and Gamma Distributions

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Proof: MGF of Gamma Distribution

The density is

λ^α

f_X(x) =

x^α−1e^−λx

,

x > 0.

Γ(α)

By deﬁnition,

Z

∞

M_X(t) = E(e^tX) =

e^txf_X(x) dx.

0

Therefore,

Z

∞

λ^α

M_X(t) =

x^α−1e^txe^−λxdx.

Γ(α)

0

Combining the exponentials, we have

Z

∞

λ^α

M_X(t) =

x^α−1

e

^−(λ−t)xdx.

Γ(α)

0

Using the Gamma integral,

Z

∞

Γ(α)

c^α

x^α−1

e

^−cxdx =

,

c > 0,

0

with c = λ − t, we get

ꢀ

Chapter 8B: Beta and Gamma Distributions

ꢁ

α

λ

M_X(t) =

,

for λ − t > 0.

λ − t

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Relationship Between Beta and Gamma

If

X ∼ Gamma(a, 1),

Y ∼ Gamma(b, 1),

∼ Beta(a, b).

independently, then

X

X + Y

Beautiful connection

Beta and Gamma distributions are deeply linked.

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Summary

▶ Beta distributions model random quantities between

0

and 1.

▶ Gamma functions generalize factorials.

▶ Gamma distributions generalize Exponential distributions.

▶ Gamma distributions arise from sums of independent Exponentials.

▶ Beta and Gamma distributions are closely connected.

Next lecture

Conditional expectation and prediction.

Arman Jahangiri

Chapter 8B: Beta and Gamma Distributions

Summer 2026 27