Exercise 2.16 - Mean, mode, variance for the beta distribution

Answers

Firstly, we derive the mode for beta distribution by differentiating the pdf:

\frac{d}{d x} x^{a - 1} {(1 - x)}^{b - 1} = [(1 - x) (a - 1) - (b - 1) x] x^{a - 2} {(1 - x)}^{b - 2} .

Setting this to zero yields:

mode = \frac{a - 1}{a + b - 2} .

Secondly, derive the moment in beta distribution:

\begin{align} 𝔼 [x^{N}] = & \frac{1}{B (a, b)} \int x^{a + N - 1} {(1 - x)}^{b - 1} 𝑑𝑥 \\ = & \frac{B (a + N, b)}{B (a, b)} \\ = & \frac{Γ (a + N) Γ (b)}{Γ (a + N + b)} \frac{Γ (a + b)}{Γ (a) Γ (b)} . \end{align}

Setting $N = 1, 2$ :

𝔼 [x] = \frac{a}{a + b},

𝔼 [x^{2}] = \frac{a (a + 1)}{(a + b) (a + b + 1)} .

Where we have used the properties of the Gamma function. Finally:

mean = 𝔼 [x] = \frac{a}{a + b},

variance = 𝔼 [x^{2}] - 𝔼^{2} [x] = \frac{𝑎𝑏}{{(a + b)}^{2} (a + b + 1)} .

solour_lfq

2021-03-24 13:42