2.15.1 Beta

Question 1

What is the support of the beta distribution?

Answer 1

The beta distribution is defined on the interval [0, 1].

Question 2

How do you write the beta distribution with parameters?

Answer 2

If X ~ Beta(α, β), then X is a beta random variable with shape parameters α > 0 and β > 0.

Question 3

What is the PDF of the beta distribution?

Answer 3

f(x) = [x^(α - 1) * (1 - x)^(β - 1)] / B(α, β), for 0 < x < 1

Question 4

What is the beta function B(α, β)?

Answer 4

B(α, β) = ∫₀¹ t^(α - 1) * (1 - t)^(β - 1) dt

Question 5

What happens when α = β = 1?

Answer 5

The beta distribution becomes Uniform(0, 1) with f(x) = 1 for 0 < x < 1.

Question 6

Does the beta distribution have a closed-form CDF?

Answer 6

No — the CDF must be computed numerically.

Question 7

What is the mean of X ~ Beta(α, β)?

Answer 7

E[X] = α / (α + β).

Question 8

What is the variance of X ~ Beta(α, β)?

Answer 8

Var(X) = (α * β) / [(α + β)² * (α + β + 1)].

Question 9

What shapes can the beta distribution take?

Answer 9

Uniform, U-shaped, J-shaped, bell-shaped — it’s super flexible.

Question 10

What does the beta distribution model in practice?

Answer 10

Proportions and probabilities — anything bounded between 0 and 1.

Question 11

How do you find the constant in a beta-like PDF?

Answer 11

Integrate the PDF over [0, 1] and set it equal to 1 to solve for the constant.

Question 12

How do you recognize a beta-shaped PDF?

Answer 12

It has the form: f(x) = k * x^(α - 1) * (1 - x)^(β - 1), on [0, 1].

Question 13

What is the shortcut to compute Var(X) if you know E[X] and E[X²]?

Answer 13

Var(X) = E[X²] - (E[X])².