Moment Generating Function of Gamma Distribution, Harmonic Mean and Mode of Gamma Distribution

Let X follow a gamma distribution with the parameterα i.e. X ~ G(α). The moment generating function of gamma distribution is given by

$$M_{x}(t) \ = \ E({e^t}^{x})$$

$$= \ \int_{0}^{\infty} \ {e^t}^{x} \ f(x) \ dx \ = \ \int_{0}^{\infty} \ {e^t}^{x} \ \frac{{e^-}^{x} \ {x^\alpha}^{-1}}{\Gamma\alpha} \ dx$$

$$= \ \frac{1}{\Gamma\alpha} \ \int_{0}^{\infty} \ {e^-}^{(1-t)x} \ {x^\alpha}^{x-1} \ dx$$

$$= \ \frac{1}{\Gamma\alpha} \ \int_{0}^{\infty} \ {e^-}^{y} \ { \left ( \ \frac{y}{1-t} \ \right ) ^\alpha}^{-1} \ \frac{dy}{1-t}$$

$$Where, \ y \ = \ (1-t) \ x \ \Rightarrow \ dy \ = \ (1-t) \ dx$$

$$= \ \frac{1}{\Gamma\alpha} \ \frac{1}{(1-t)^\alpha} \ \int_{o}^{\infty} \ {e^-}^{y} \ {y^\alpha}^{-1} \ dy$$

$$= \ \frac{1}{\Gamma\alpha} \ \frac{1}{(1-t)^\alpha} \ \Gamma\alpha$$

$$= \ \frac{1}{(1-t)^\alpha}$$

$$\Rightarrow M_X(t) \ = \ {(1-t)^-}^{\alpha} \ \ \ \ ; \ | t | \ < \ 1$$

Now, the moments of gamma distribution can be determined by expanding the moment generating function M_x(t).

ExpandingM_x(t), we have

$$M_{x}(t) \ = \ 1 \ + \ \alpha \ + \\binom{\alpha + 1}{2} \ t^2 \ + \ . \ . \ . \ + \ \binom{\alpha + r - 1}{r} \ t^r \ + \ . \ . \ .$$

$$\therefore \ \mu_{r}^{'} \ = \ Coefficient \ of \ \frac{t^r}{r!}$$

$$= \ \frac{(\alpha \ + \ r \ - \ 1)!}{(\alpha \ - \ 1)!}$$

$$\mu_{r}^{'} \ = \ \frac{\Gamma(\alpha \ + \ r)}{\Gamma\alpha} \ \ \ \ ; \ r \ = \ 1, \ 2, \ 3, \ 4$$

Then putting r = 1, 2, 3, 4, the first four moments about origin can be determined and then central moments of the gamma distribution can be determined.

It is noted that the moments of the gamma distribution can also be found by using the derivatives of M_x(t) at t = 0. i.e.

$$\mu_{r}^{'} \ = \ \frac{d^r}{dt^r} \ M_{x}(t) \mid_{t=0}$$

$$= \ \frac{d^r}{dt^r} \ {(1-t)^-}^\alpha \ \mid_{t=0}$$

$$\therefore \ \mu_{r}^{'} \ = \ \frac{d}{dt} \ {(1-t)^-}^\alpha \ \mid_{t=0}$$

$$= \ - \ \alpha \ {(1-t)^-}^\alpha \ (-1) \ \mid_{t=0}$$

$$ = \ \alpha$$

$$\Rightarrow \ Mean \ = \ \alpha.$$

$$\mu_{2}^{'} \ = \ \frac{d^2}{dt^2} \ {(1-t)^-}^\alpha \ \mid_{t=0}$$

$$= \ \alpha \ (- \ \alpha \ - \ 1) \ {(1 \ - \ t)^-}^{\alpha \ - \ 2} \ (- \ 1) \ \mid_{t=0}$$

$$= \ \alpha \ (\alpha \ + \ 1)$$

$$\mu_{2} \ = \ \mu_{2}^{'} \ - \ (\mu_{1}^{'})^2 \ = \ \alpha \ (\alpha \ + \ 1) \ - \ {\alpha}^2 \ = \ \alpha.$$

$$\Rightarrow \ Variance \ = \ \alpha.$$

Harmonic mean and mode of Gamma distribution

We have gamma distribution G(α) has the probability density function (pdf)

$$f(x) \ = \ \frac{{e^-}^{x} \ {x^\alpha}^{-1}}{\Gamma\alpha} \ \ \ \ ; \ 0 \ < x \ < \ \infty$$

The harmonic mean H of the gamma distribution is given by

$$\frac{1}{H} = \ E \ \left ( \frac{1}{X} \right )$$

$$= \ \int_{0}^{\infty} \ \frac{1}{x} \ f(x) \ dx$$

$$= \ \frac{1}{\Gamma\alpha} \ \int_{0}^{\infty} \ \frac{1}{x} \ {e^-}^{-x}. \ {x^\alpha}^{-1} \ dx$$

$$= \ \frac{1}{\Gamma\alpha} \ \int_{0}^{\infty} \ {e^-}^{x}. \ {x^\alpha}^{-2} \ dx$$

$$= \ \frac{\Gamma \ (\alpha \ - \ 1)}{\Gamma\alpha} \ = \ \frac{(\alpha - 2)!}{(\alpha - 1)!} \ = \ \frac{1}{\alpha - 1}$$

$$\therefore \ H \ = \ \alpha \ - \ 1$$

The mode of the gamma distribution is given by the solution of f^' (x) = 0 and f ^'' (x) < 0.

Thus,

$$f^{' }(x) \ = \ 0$$

$$or, \ \ \frac{d}{dx} \ \left [ \frac{{e^-}^{x} \ {x^\alpha}^{-1}}{\Gamma\alpha} \ \right ] \ = \ 0$$

$$or, \ \ \frac{1}{\Gamma\alpha} \ [ \ {e^-}^{x} \ {x^\alpha}^{-1} \ + \ (\alpha \ - \ 1) \ {e^-}^{x} \ {x^\alpha}^{-2} \ ] \ = \ 0$$

$$0r, \ \ \frac{{e^-}^{x} \ {x^\alpha}^{-1}}{\Gamma\alpha} \ \left [ \ 1 \ - \ \frac{(\alpha \ - \ 1 \ )}{x} \ \right ] \ = \ 0$$

$$or, \ \ \ f(x) \ \left [ \frac{x - (\alpha - 1)}{x}\ \right ] \ = \ 0$$

$$or, \ \ \ \frac{x-(\alpha - 1)}{x} \ = \ 0$$

$$\Rightarrow \ x \ = \ \alpha \ - \ 1$$

Also, it can be proved that f '' (x) < 0 at x = α - 1. Hence, the mode of the gamma distribution isα - 1 forα > 1. It is noted that ifα < 1, the mode is zero.

Additive Property of Gamma Distribution

The sum of independent gamma varities is also a gamma varietes. That is, If X₁, X₂, ..., Xn are n independent gamma varieties with parametersα₁,α₂, . . .,α_n respectively, then X₁+ X₂ + . . . + X_n. is also a gamma variate with parameterα₁ +α₂ + . . . +α_n.

Bibliography

Sukubhattu N.P. (2013). Probability & Inference - II. Asmita Books Publishers & Distributors (P) Ltd., Kathmandu.

Larson H.J. Introduction to Probability Theory and Statistical Inference. WileyInternational, New York.