Basic Statistics

Question 1

Give the Gamma function

Answer 1

Γ(α) = integral (0 to inf) x^(α−1)e^(−x) dx, α > 0

Question 2

Give the Beta function

Answer 2

B(α, β) = integral(0 to 1)

x^(α−1) * (1 − x)^(β−1) dx α, β > 0

Question 3

What properties do the Gamma and Beta functions satisfy?

Answer 3

1. Γ(α + 1) = αΓ(α)
2. If n ∈ N then Γ(n + 1) = n!
3. B(α, β) = Γ(α)Γ(β) / Γ(α+β)

Question 4

What density function does a Gamma random variable have?

Answer 4

fx(x) = (b^a / Γ(a)) * x^(a−1) * e^(−bx) , x > 0

Question 5

What density function does a Beta random variable have?

Answer 5

fy(y) = (1/B(a, b)) * y^(a−1) * (1 − y)^(b−1) , 0 < y < 1

Question 6

Mean of a Gamma variable?

Answer 6

E(X) = a/b

Question 7

Mean of a Beta variable?

Answer 7

E(Y) = a / (a+b)