Weibull Distribution

1. Weibull Probability Plot
Light bulbs are tested for both lamp life and strength, selected bulbs
are screwed into life test racks and lit at levels far exceeding their
normal burning strength. This produces an accurate reading on how
long the bulb will last under normal conditions. The average life of
the majority of household light bulbs is 750 to 1000 hours, depending
on wattage. Determining a light bulbs life span at best is very difficult,
performance data is not generally available and costly to generate
since large numbers of light bulbs must be tested to destruction. We
the engineers at the company want to guarantee the bulbs for 10 years
of operation. First we must stress the bulbs to simulate long-term use
and record the hours until failure for each bulb.





















Figure 2 – Weibull Probability Plot
Reading the Graph
To determine the ?????? and β parameters for failures of a light bulb within
a given population, it is necessary to conduct a life test on a small
selection sample of units. The percent of the sample failing is then
plotted against the time of failure, on Weibull probability paper. The
characteristic life ?????? of the population is defined as the time when
63.2% of the population has failed and this is obtained directly from
the graph. The slope β of the graph is given by drawing a parallel line
on the β scale outlined on the graph and corresponds to the shape
factor of the distribution.

Conclusions
Don't expect the majority of bulbs to survive past the characteristic
life. In order to achieve an acceptably low number of bulb failures
within a specific period, the characteristic life of the lights must be
much longer than the desired lifetime. If we make claims about our
product lifetimes, we must have corroborating evidence to prove our
claims.


WEIBULL DISTRIBUTION
STUDENT: CIARAN NOLAN
LECTURER: DR RAURÍ MCCOOL
Introduction
Waloddi Weibull was born on June 18, 1887 in Sweden. The probability
distribution named after him was studied in his paper 'A Statistical
Distribution Function of Wide Applicability‘. The Weibull distributions
is immensely popular in reliability, because it includes distributions of
decreasing, constant, and increasing failure rates. The main advantage
of Weibull analysis is that its able to detect accurate failure rates with
extremely small samples.

The bathtub curve is widely used in reliability engineering. It describes
a particular form of the hazard function which comprises three parts:
1. Decreasing failure rate, known as early failures.
2. Constant failure rate, known as random failures.
3. Increasing failure rate, known as wear-out failures.









Figure 1 – The Bathtub Curve Hazard Function
In reliability, the cumulative distribution function corresponding to a
bathtub curve may be analyzed using a Weibull Probability plot.
Problem Objective
The key objective is to develop an operational poster for the shop floor
so staff can easily understand the application and use of the Weibull
method.
Method
The most general expressions of the Weibull is given by the Weibull
Reliability function:
R(t) = exp −
??????−t0
??????
??????

The Weibull Instantaneous Failure Rate or Hazard Rate, which is
expressed by:
???????????? =
??????
??????
??????−t0
??????
??????−1

• β is the shape parameter
• η is the scale parameter
• t0 is the location parameter
Often the location parameter t0 is not used, so the value can be zero. β
the Weibull shape parameter is equal to the slope of the line in the
probability plot. The scale parameter η defines where the bulk of the
distribution lies.

The slope β, also indicates which class of failures is present:
β = 1.0 Means random failures, Exponential distribution.
β < 1.0 Indicates infant mortality, Decreasing failure rate.
β > 1.0 Indicates wear out failures, Increasing failure rate.