The mean of gamma distribution is the expected value \(E[X]\) of a continuous random variable \(X\) whose probability density function follows a gamma form. Two parameterizations are common (shape-scale and shape-rate), and the mean depends on which one is used.
For shape \(k>0\) and scale \(\theta>0\), the gamma density is \[ f(x)=\frac{1}{\Gamma(k)\,\theta^{k}}\,x^{k-1}e^{-x/\theta},\quad x>0. \] The mean of gamma distribution in this form is \[ E[X]=k\cdot\theta. \]
Equivalent parameterization (shape-rate)
Some texts use shape \(\alpha>0\) and rate \(\beta>0\), where \(\theta=1/\beta\). The density is \[ f(x)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}\,x^{\alpha-1}e^{-\beta x},\quad x>0, \] and the mean of gamma distribution becomes \[ E[X]=\frac{\alpha}{\beta}. \]
| Parameterization | Parameters | PDF form | Mean | Variance (often paired with the mean) |
|---|---|---|---|---|
| Shape-scale | \(k,\theta\) | \(\dfrac{1}{\Gamma(k)\theta^k}x^{k-1}e^{-x/\theta}\) | \(E[X]=k\cdot\theta\) | \(\mathrm{Var}(X)=k\cdot\theta^2\) |
| Shape-rate | \(\alpha,\beta\) | \(\dfrac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}\) | \(E[X]=\alpha/\beta\) | \(\mathrm{Var}(X)=\alpha/\beta^2\) |
Derivation of the mean in the shape-scale form
The expected value of a continuous random variable is \(E[X]=\int_0^\infty x f(x)\,dx\). Using the shape-scale gamma density:
- Substitute the density into the expectation integral: \[ E[X]=\int_{0}^{\infty}x\cdot\frac{1}{\Gamma(k)\theta^{k}}x^{k-1}e^{-x/\theta}\,dx =\frac{1}{\Gamma(k)\theta^{k}}\int_{0}^{\infty}x^{k}e^{-x/\theta}\,dx. \]
- Use the substitution \(u=x/\theta\), so \(x=\theta u\) and \(dx=\theta\,du\): \[ \int_{0}^{\infty}x^{k}e^{-x/\theta}\,dx =\int_{0}^{\infty}(\theta u)^{k}e^{-u}\,\theta\,du =\theta^{k+1}\int_{0}^{\infty}u^{k}e^{-u}\,du. \]
- Recognize the gamma function \(\Gamma(k+1)=\int_{0}^{\infty}u^{k}e^{-u}\,du\): \[ E[X]=\frac{1}{\Gamma(k)\theta^{k}}\cdot\theta^{k+1}\Gamma(k+1) =\theta\cdot\frac{\Gamma(k+1)}{\Gamma(k)}. \]
- Apply the recursion \(\Gamma(k+1)=k\Gamma(k)\): \[ E[X]=\theta\cdot\frac{k\Gamma(k)}{\Gamma(k)}=k\theta. \]
Numeric example
Suppose \(X\sim\mathrm{Gamma}(k=3,\theta=2)\) in the shape-scale parameterization. Then the mean of gamma distribution is \[ E[X]=k\theta=3\cdot 2=6. \] Using the shape-rate form gives \(\beta=1/\theta=0.5\) and \(\alpha=k=3\), so \(E[X]=\alpha/\beta=3/0.5=6\), matching the same mean.
Visualization: gamma density with the mean marked
Common pitfalls when using the mean of gamma distribution
- Scale vs rate confusion: \(\theta\) is the scale, \(\beta\) is the rate, and \(\theta=1/\beta\). The mean is \(k\theta\) or \(\alpha/\beta\), depending on notation.
- Skewness: For many parameter choices, the gamma distribution is right-skewed; the mean can exceed the median, so “typical” values may be smaller than the mean.
- Units: The mean has the same units as \(X\). If \(X\) measures time, then \(E[X]\) is a time.
The mean of gamma distribution equals shape times scale: \(E[X]=k\theta\), equivalently shape divided by rate: \(E[X]=\alpha/\beta\).