Report

Part 1

1.1 What is the probability of surviving past 95 years given a constant failure rate?

The probability of surviving past 95 years is 0.51%.

1.2 What is the probability of surviving past 95 years given the Weibull failure rate?

The probability of surviving past 95 years is 0.15%.

Rubric: - 0.5 points in total

0.5 point for computing the correct probability (also award this point for a different probability, if the group got a different result as a consequence of the scale error mentioned in the erratum)

Part 2

2.1 Describe the density function plotted in Task 4. What does it represent? What patterns can you glean from it?

The density function describes the probability density of dying at a given age. We can see that mortality is very high among infants, drops significantly in childhood, adolescence, and maturity, and grows significantly with age. Mortality drops again towards 100 years of age as most people die before that.

Rubric: - 1.5 points in total

0.5 point for recognizing that mortality is high in early life 0.5 point for recognizing that mortality lowers in-between (childhood, adolescence, maturity) 0.5 point for recognizing that mortality is high in late life

2.2 Discuss the hazard rate functions for the constant failure rate model and the Weibull model in comparison with the empirical (from data) estimation. What do you observe a) in the beginning of life and b) by the end of life?

In the constant failure rate model, failure rates are (surprise, surprise) constant at all times. In the Weilbull model, failure rates are initially very low but increase until they plateau at a fixed value. In the empirical model, we see that failure rates are initially high (infant mortality), then reach a plateau of stability during childhood and maturity, and eventually increase again in old age, resulting in a “bathtub” curve.

Rubric: - 2.5 points in total

0.5 point for stating that constant failure rate is constant, or something equivalent 0.5 points for stating that Weibull failure rates start low 0.5 points for stating that Weibull failure rates end at a plateau, or something equivalent 0.5 points for stating that the empirical failure rates are high in early and late life 0.5 points for stating that the empirical failure rates are low in-between

2.3 Discuss the reliability functions for the constant failure rate model and the Weibull model in comparison with the empirical (from data) estimation.

The reliability (or survival probability) function represents the probability of surviving up to a given age. As we can see, both the constant and Weibull models dramatically overestimate the mortality in youth and maturity, predicting that about 80% of the population would have died at age 30. By contrast, the empirical reliability function maintains a high plateau and then drops sharply past an age of about 50.

Rubric: - 1 points in total

0.5 point for recognizing that the constant and Weibull reliability function drop fast in early age, then flatten out 0.5 point for recognizing that the empirical reliability function remains high for longer, then drops fast in older age

Part 3

3.1 How should life insurance be priced by the company under the three different models? Consider insurance policies that change their premiums with age.

Real life insurances do of course have a more complex pricing scheme, but a simple approach would be to scale the insurance pricing with the risk of at any given age. For the constant failure model, insurance premiums would be constant with age and would require no adjustment. In the Weibull model, insurance premiums would be cheap in infancy and then level out at a constant price. In the empirical model, failure rates are high in infancy and old age, which should command correspondingly higher premiums, whereas mortality is low in childhood, adolescence, and maturity, which should result in cheap insurance premiums.

Rubric: - 1.5 points in total

the solutions should be based on the instant probability of failure. If students interpreted this question to use a more complex but reasonable pricing scheme, also award points.

0.5 point for a “constant” scheme that has constant pricing 0.5 point for a “weibull” scheme that starts cheap but then plateaus at a certain price 0.5 point for an “empirical” scheme that has starts moderately expensive, then becomes cheaper, then becomes very expensive

3.2 Think of other examples in engineering or everyday life and discuss what failure curves these examples follow. Explain why. Extra points for finding engineering examples of “bathtub” curves.

Constant failure rates usually occur in (statistically) stationary memory-less processes, i.e., failures that just randomly occur without any warning. For instance, we might assume that earthquakes above a certain magnitude follow a constant failure rate, if we neglect aftershocks. Other examples might include random sewer blockage, being struck by lightning, or winning a lottery ticket.

The Weibull or Gamma distributions can have either a rising or falling failure rate, see the interactive element in the book. Processes with a rising failure rate are quite abundant and could include processes in which material fatigue or cumulate damages increase the chance of failure. Examples might be steel bridges rusting, asphalt pavements cracking, or soil slopes undergoing erosion. Processes with a falling failure rate are more rare and represent processes that reinforce and strengthen themselves with time. Examples might include Roman concrete, which hardens with time, or sloped highway embankments, which become more resistant to erosion after the root systems of vegetation start stabilizing the slopes.

Bathtub failure rates describe processes that are vulnerable early and late, but stable in-between. Examples might include concrete constructions, which are vulnerable before the concrete hardens, then remain stable for a long time, and become vulnerable again as the material ages. Another example might be wastewater treatment plans, which can suffer from distruptions as the plant starts operation, and then again later in age as the installations age and become more prone to material failure.

Rubric: - 2 (3 with bonus) points in total

1 point for a reasonable example 1 point for another reasonable example 1 bonus point (up to the point maximum) if one of these examples is a bathtub curve

By Max Ramgraber, Renan Barros and Oswaldo Morales Napoles, Delft University of Technology. CC BY 4.0, more info on the Credits page of Workbook