Unit 3 — Gamma & Beta Distributions
Gamma — definition, moments, MGF, additive property, limiting form. Beta of first & second kind — definition, mean, variance, harmonic mean.
Topics Covered
Gamma Function
Gamma Distribution
Additive Property
Limiting Form
Beta of First Kind
Beta of Second Kind
Harmonic Mean
On this page
1. Preliminaries: Gamma Function 2. Gamma Distribution 3. Beta Distribution of First Kind 4. Beta Distribution of Second Kind Key Take-aways
1. Preliminaries: Gamma Function
DEFINITION
For \(\alpha > 0\), the Gamma function is
\[
\Gamma(\alpha) \;=\; \int_0^\infty x^{\alpha - 1} e^{-x}\,dx.
\]
Properties:
\(\Gamma(\alpha + 1) = \alpha\,\Gamma(\alpha)\) (recurrence).
\(\Gamma(n) = (n - 1)!\) for positive integer \(n\).
\(\Gamma(1/2) = \sqrt{\pi}\).
2. Gamma Distribution
DEFINITION
A continuous r.v. \(X\) follows a Gamma distribution with parameters
\(\alpha > 0\) (shape) and \(\lambda > 0\) (rate) if its PDF is
\[
f(x) \;=\; \dfrac{\lambda^\alpha}{\Gamma(\alpha)}\, x^{\alpha - 1}\, e^{-\lambda x}, \quad x \ge 0.
\]
Notation: \(X \sim G(\alpha, \lambda)\). When \(\alpha = 1\), Gamma reduces to Exp\((\lambda)\).
Moments & MGF
MGF
\[
M_X(t) \;=\; \left(\dfrac{\lambda}{\lambda - t}\right)^\alpha, \quad t < \lambda.
\]
\[
E(X) = \dfrac{\alpha}{\lambda}, \qquad
\text{Var}(X) = \dfrac{\alpha}{\lambda^2}.
\]
Higher raw moments: \(\mu'_r = \dfrac{\alpha(\alpha+1)\cdots(\alpha+r-1)}{\lambda^r} = \dfrac{\Gamma(\alpha+r)}{\lambda^r\, \Gamma(\alpha)}\).
CF and CGF
\[
\phi_X(t) = \left(\dfrac{\lambda}{\lambda - it}\right)^{\alpha}, \qquad
K_X(t) = \alpha\bigl[\ln \lambda - \ln(\lambda - t)\bigr].
\]
Skewness and Kurtosis
\[
\gamma_1 = \dfrac{2}{\sqrt{\alpha}}, \qquad
\beta_2 = 3 + \dfrac{6}{\alpha}, \qquad
\gamma_2 = \dfrac{6}{\alpha}.
\]
Always right-skewed and leptokurtic; both decrease as \(\alpha\) grows.
Additive Property
Hence sum of \(n\) i.i.d. Exp\((\lambda)\) variables is \(G(n, \lambda)\).
α=1 (Exponential)
α=2
α=5
0
Gamma PDF for various shape α (λ=1)
Fig 3.1 — Larger α → more bell-like
By CLT, for large \(\alpha\), the standardized Gamma converges to standard normal:
\[
\dfrac{X - \alpha/\lambda}{\sqrt{\alpha}/\lambda} \;\xrightarrow{d}\; N(0, 1).
\]
Worked Examples
EXAMPLE 1
The waiting time for the third success in a Poisson process with rate \(\lambda = 2\) per hour
follows \(G(3, 2)\). Mean wait = 3/2 = 1.5 h; Variance = 3/4.
EXAMPLE 2 (Probability)
For \(X \sim G(2, 1)\), find \(P(X > 3)\).
\(f(x) = x e^{-x}\). \(P(X > 3) = \int_3^\infty x e^{-x}\,dx = (3+1)e^{-3} = 4 e^{-3} \approx 0.1991\).
3. Beta Distribution of First Kind
DEFINITION
A continuous r.v. \(X\) follows a Beta distribution of the first kind with
parameters \(\alpha, \beta > 0\) if its PDF is
\[
f(x) \;=\; \dfrac{1}{B(\alpha, \beta)}\, x^{\alpha-1}\, (1-x)^{\beta-1}, \quad 0 \le x \le 1.
\]
Where the Beta function \(B(\alpha, \beta) = \dfrac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}\).
Notation: \(X \sim \beta_1(\alpha, \beta)\).
Mean, Variance & Harmonic Mean
\[
E(X) = \dfrac{\alpha}{\alpha + \beta}, \qquad
\text{Var}(X) = \dfrac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}.
\]
\[
\text{HM} = \dfrac{\alpha - 1}{\alpha + \beta - 1}, \quad \alpha > 1.
\]
Special cases
\(\alpha = \beta = 1\): Uniform(0,1).
\(\alpha = \beta\): symmetric about 0.5.
\(\alpha = \beta = 1/2\): Arcsine distribution.
Beta(α, β): one family, many shapes on [0, 1]
0
0.25
0.5
0.75
1
0
1
2
x
density
Beta(1,1) = Uniform
Beta(½,½) arcsine (U-shape)
Beta(2,2) symmetric
Beta(2,5) right-skewed
Beta(5,2) left-skewed
Fig 3.2 — The Beta family is remarkably flexible: the same density \(f(x)=x^{\alpha-1}(1-x)^{\beta-1}/B(\alpha,\beta)\) is flat when \(\alpha=\beta=1\) (uniform), U-shaped when \(\alpha=\beta=\tfrac12\) (arcsine), a symmetric bell when \(\alpha=\beta>1\), and skewed toward the larger parameter otherwise (mode at \((\alpha-1)/(\alpha+\beta-2)\)). This is why Beta is the natural model for a proportion on \([0,1]\).
EXAMPLE 1
For \(\beta_1(2, 3)\): PDF \(f(x) = 12x(1-x)^2,\; 0 \le x \le 1\).
EXAMPLE 2 (Probability)
For \(\beta_1(2, 2)\): \(f(x) = 6x(1-x)\). \(P(X \le 0.5) = \int_0^{0.5} 6x(1-x)\,dx = 6[x^2/2 - x^3/3]_0^{0.5} = 6(0.125 - 0.0417) = 0.5\).
4. Beta Distribution of Second Kind
DEFINITION
A continuous r.v. \(Y\) follows a Beta distribution of the second kind with parameters
\(\alpha, \beta > 0\) if its PDF is
\[
f(y) \;=\; \dfrac{1}{B(\alpha, \beta)} \cdot \dfrac{y^{\alpha - 1}}{(1 + y)^{\alpha + \beta}}, \quad y \ge 0.
\]
Notation: \(Y \sim \beta_2(\alpha, \beta)\). It can be obtained from Beta-I by the transformation \(Y = X/(1-X)\).
Mean, Variance & Harmonic Mean
\[
E(Y) = \dfrac{\alpha}{\beta - 1}, \quad \beta > 1.
\]
\[
\text{Var}(Y) = \dfrac{\alpha(\alpha + \beta - 1)}{(\beta - 1)^2(\beta - 2)}, \quad \beta > 2.
\]
\[
\text{HM} = \dfrac{\alpha - 1}{\beta}, \quad \alpha > 1.
\]
EXAMPLE 1
For \(\beta_2(2, 4)\): Mean \(= 2/3 \approx 0.667\); Variance \(= 2 \cdot 5 / (9 \cdot 2) = 0.556\).
EXAMPLE 2
For \(\beta_2(3, 5)\) — \(\alpha > 1, \beta > 2\): HM \(= 2/5 = 0.4\); Mean \(= 3/4 = 0.75\); Variance \(= 3 \cdot 7/(16 \cdot 3) = 0.4375\).
Key Take-aways
Gamma\((\alpha, \lambda)\): \(f(x) = \lambda^\alpha x^{\alpha-1} e^{-\lambda x}/\Gamma(\alpha)\); mean \(\alpha/\lambda\); var \(\alpha/\lambda^2\).
MGF \((\lambda/(\lambda-t))^\alpha\); additive over \(\alpha\) with same \(\lambda\); → Normal as \(\alpha\to\infty\).
Beta-I \((\alpha,\beta)\) on [0,1]: mean \(\alpha/(\alpha+\beta)\); HM \((\alpha-1)/(\alpha+\beta-1)\).
Beta-II on \([0,\infty)\): mean \(\alpha/(\beta-1)\); HM \((\alpha-1)/\beta\).
\(\alpha=\beta=1\) Beta-I = Uniform(0,1).
Unit 3 — Gamma & Beta Distributions