Suppose X has a gamma distribution. Prove that its moments...

Suppose $X$ has a gamma distribution. Prove that its moments are given by $$ E\left[X^n\right]=\left[\prod_{k=1}^{n-1}\left(1+k C_X^2\right)\right] E[X]^n \quad n=2,3, \cdots . $$ Since the squared coefficient of variation is given by $$ C_X^2=\frac{1}{\beta}, $$ this means that we can write the above formula as $$ E\left[X^n\right]=\frac{\beta(\beta+1)(\beta+2) \cdots(\beta+n-1)}{\alpha^n}, \quad n=1,2,3, \cdots . $$ This result also implies that for the Erlang- $k$ random variable, we have $$ E\left[X^n\right]=\left(1+\frac{1}{k}\right)\left(1+\frac{2}{k}\right) \cdots\left(1+\frac{n-1}{k}\right) E[X]^n, \quad n=1,2,3, \cdots . $$

Suppose $X$ has a gamma distribution. Prove that its moments are given by
$$
E\left[X^n\right]=\left[\prod_{k=1}^{n-1}\left(1+k C_X^2\right)\right] E[X]^n \quad n=2,3, \cdots .
$$
Since the squared coefficient of variation is given by
$$
C_X^2=\frac{1}{\beta},
$$
this means that we can write the above formula as
$$
E\left[X^n\right]=\frac{\beta(\beta+1)(\beta+2) \cdots(\beta+n-1)}{\alpha^n}, \quad n=1,2,3, \cdots .
$$
This result also implies that for the Erlang- $k$ random variable, we have
$$
E\left[X^n\right]=\left(1+\frac{1}{k}\right)\left(1+\frac{2}{k}\right) \cdots\left(1+\frac{n-1}{k}\right) E[X]^n, \quad n=1,2,3, \cdots .
$$

Key Concepts

Erlang Distribution

The Erlang distribution is a special case of the gamma distribution where the shape parameter is an integer. It is extensively used in fields such as queueing theory and reliability engineering. The moment formulas for the Erlang distribution simplify in a similar product form as those for the gamma distribution, illustrating the structural similarity between these two distributions while highlighting the practicality and interpretability when the shape parameter is an integer.

Coefficient of Variation

The coefficient of variation (CV) is a normalized measure of dispersion defined as the ratio of the standard deviation to the mean. In many distributions, including the gamma distribution, the squared coefficient of variation (CV²) is of interest because it relates variance to the square of the mean, providing a scale-invariant measure of spread that is directly incorporated into moment formulas.

Gamma Distribution

The gamma distribution is a two-parameter family of continuous probability distributions commonly used to model waiting times and processes with skewed distributions. Its probability density function is characterized by a shape parameter and a scale or rate parameter. An important property of the gamma distribution is that its moments can be computed exactly using the gamma function, which connects the distribution’s parameters to its expected values and higher order moments.

Moments of a Distribution

Moments are quantitative measures related to the shape of a probability distribution. They provide insight into properties such as the mean (first moment) and variance (second central moment). In the context of the gamma distribution, the nth moment can be derived analytically using a product formula, showing a systematic relationship between the parameters and the moment order.

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