GAMMA

Application:

Calculating the Probability of a Capacitor's Lifetime

Suppose the lifetime of a capacitor is modeled by a Gamma distribution with a shape parameter α=3 and a rate parameter β=0.5. We want to calculate the probability density at a specific lifetime, say x=5 years.

First, we need to calculate the value of the Gamma function for α=3, which is $\Gamma$(3). Since 3 is an integer, $\Gamma$(3)=(3−1)!=2!=2.

Now we can substitute the values into the PDF equation:

$f(5;3,0.5)=\frac{(0.5{)}^3}{\Gamma (3)}(5{)}^{3-1}{e}^{-(0.5)(5)}$

$f(5;3,0.5)=\frac{0.125}{2}(5{)}^2{e}^{-2.5}$

$f(5;3,0.5)=0.0625\times 25\times e^{-2.5}$

$f(5;3,0.5)=1.5625\times 0.082085$

$f(5;3,0.5)\approx 0.128$

This value, 0.128, represents the probability density at a lifetime of 5 years.

The Role of the Gamma Function in the Calculation

The Gamma function is essential for calculating the normalization constant, $\frac{{\beta }^{\alpha }}{\Gamma (\alpha )}$, which ensures that the total area under the probability density curve is exactly 1. Without this normalization, the function would not be a valid probability distribution.

The following table shows how the value of the Gamma function changes with its parameter x: