Template:Example: Standard Actuarial Example

Standard Actuarial Example

Find reliability estimates for the data in the Simple-Actuarial Example using the standard actuarial method.

Solution

The solution to this example is similar to that of Simple-Actuarial Example, with the exception of the inclusion of the [math]\displaystyle{ n_{i}^{\prime } }[/math] term, which is used in equation for the standard actuarial method. Applying this equation to the data, we can generate the following table:

[math]\displaystyle{ \begin{matrix} Start & End & Number of & Number of & Adjusted & {} & {} \\ Time & Time & Failures, {{r}_{i}} & Suspensions, {{s}_{i}} & Units, n_{i}^{\prime } & 1-\tfrac{{{r}_{j}}}{n_{j}^{\prime }} & \prod\mathop{}_{}^{}1-\tfrac{{{r}_{j}}}{n_{j}^{\prime }} \\ 0 & 50 & 2 & 4 & 53 & 0.962 & 0.962 \\ 50 & 100 & 0 & 5 & 46.5 & 1.000 & 0.962 \\ 100 & 150 & 2 & 2 & 43 & 0.953 & 0.918 \\ 150 & 200 & 3 & 5 & 37.5 & 0.920 & 0.844 \\ 200 & 250 & 2 & 1 & 31.5 & 0.937 & 0.791 \\ 250 & 300 & 1 & 2 & 28 & 0.964 & 0.762 \\ 300 & 350 & 2 & 1 & 25.5 & 0.922 & 0.702 \\ 350 & 400 & 3 & 3 & 21.5 & 0.860 & 0.604 \\ 400 & 450 & 3 & 4 & 15 & 0.800 & 0.484 \\ 450 & 500 & 1 & 2 & 9 & 0.889 & 0.430 \\ 500 & 550 & 2 & 1 & 6.5 & 0.692 & 0.298 \\ 550 & 600 & 1 & 0 & 4 & 0.750 & 0.223 \\ 600 & 650 & 2 & 1 & 2.5 & 0.200 & 0.045 \\ \end{matrix} }[/math]

As can be determined from the preceding table, the reliability estimates for the failure times are:

[math]\displaystyle{ \begin{matrix} Failure Period & Reliability \\ End Time & Estimate \\ 50 & 96.2% \\ 150 & 91.8% \\ 200 & 84.4% \\ 250 & 79.1% \\ 300 & 76.2% \\ 350 & 70.2% \\ 400 & 60.4% \\ 450 & 48.4% \\ 500 & 43.0% \\ 550 & 29.8% \\ 600 & 22.3% \\ 650 & 4.5% \\ \end{matrix} }[/math]