RANDOM EXPERIMENT AND VARIABLES A random experiment is an experiment or

a process for which the outcome cannot be predicted with certainty. The sample space (denoted S) of a random experiment is the set of all possible outcomes.

A random variable is a variable hat assumes numerical values associated with the random outcome of an experiment, where one (and only one) numerical value is assigned to each sample point.

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The sample space (denoted S) of a random experiment is the set of all possible outcomes.

An event is a subset of the sample space. We say that E has occurred if the observed outcome x is an element of E, that is x ∈ E.

Remarks: S is called the certain event. ∅ (the empty set) is called the impossible

event.

OPERATIONS ON EVENTSUnion:

a) E1 ∪ E2 occurs means E1 occurs, or E2 occurs, or both occur. b) E1 ∪ E2 ∪ · · · ∪ En occurs means that at

least one of the events E1, E2, · · · , En occurs. Intersection:

a) E1 ∩ E2 occurs means E1 occurs and E2 occurs. b) E1 ∩ E2 ∩ · · · ∩ En occurs means that all of the events E1, E2, . . . , En occur. Complement: E0 occurs means that E does not occur.

OPERATIONS ON EVENTSMutually exclusive: The events in the sequence E1, E2, . . .

are said to be mutually exclusive, if Ei ∩ Ej = ∅, for all i = j, where ∅ represents the empty set.

BASIC LIFETIME DISTRIBUTION MODELS USED FOR NON-REPAIRABLE POPULATIONS

There are a handful of parametric models that have successfully served as population models for failure times arising from a wide range of products and failure mechanisms. Sometimes there are probabilistic arguments based on the physics of the failure mode that tend to justify the choice of model. Other times the model is used solely because of its  empirical success in fitting actual failure data. 

Seven models will be described in this section:  Exponential Weibull Lognormal Gamma 

EXPONENTIAL The exponential model, with only one

unknown parameter, is the simplest of all life distribution models. The key equations for the exponential are shown below:

CONTD. Note that the failure rate reduces to the

constant λ for any time. The exponential distribution is the only

distribution to have a constant failure rate.

Also, another name for the exponential mean is the Mean Time To Fail or MTTF and we have MTTF = 1/λ.

The cumulative hazard function for the

exponential is just the integral of the failure rate or H(t)=λt.

CONTD. The PDF for the exponential has the

familiar shape shown below.

CONTD. The CDF for the exponential has the

familiar shape shown below.

USES OF THE EXPONENTIAL DISTRIBUTION MODEL

The Exponential models the flat portion of the "bathtub" curve - where most systems spend most of their "lives“.

Because of its constant failure rate property, the exponential distribution is an excellent model for the long flat "intrinsic failure" portion of the Bathtub Curve. Since most components and systems spend most of their lifetimes in this portion of the Bathtub Curve, this justifies frequent use of the exponential distribution (when early failures or wear out is not a concern).

The exponential distribution is usually used to model the time until something happens in the process.

CONTD. Just as it is often useful to approximate a curve by

piecewise straight line segments, we can approximate any failure rate curve by week-by-week or month-by-month constant rates that are the average of the actual changing rate during the respective time durations. That way we can approximate any model by piecewise exponential distribution segments patched together.

Some natural phenomena have a constant failure rate (or occurrence rate) property; for example, the arrival rate of cosmic ray alpha particles or Geiger counter tics. The exponential model works well for inter arrival times (while the Poisson distribution describes the total number of events in a given period). When these events trigger failures, the exponential life distribution model will naturally apply. 

PROBLEM TIME Calculate the exponential PDF and CDF

at 100 hours for the case where λ = 0.01.

SOLUTION:

The PDF value is 0.0037 and the CDF value is 0.6321.

WEIBULL

The Weibull is a very flexible life distribution model with two parameters. It has CDF and PDF and other key formulas given by:

α = Scale parameterγ = Shape ParameterΓ = Gamma function Γ(N)=(N−1)! for integer N.

CONTD. The cumulative hazard function for the

Weibull is the integral of the failure rate.

Special Case: When γ = 1, the Weibull reduces to the Exponential Model,

with α=1/λ = the mean time to fail (MTTF).

BATH TUB CURVE AND THE WEIBULL DISTRIBUTION

When β = 1, the hazard function is constant and therefore the data can be modelled by an exponential distribution with α=1/λ.

When β<1, we get a decreasing hazard function and

When β>1, we get a increasing hazard function

SOLUTION: Problem:A company produces automotive fuel pumps that fail according to a Weibull life distribution model with shape parameter γ = 1.5 and scale parameter 8,000 (time measured in use hours). If a typical pump is used 800 hours a year, what proportion are likely to fail within 5 years?

The probability associated with the 800*5 quantile of a Weibull distribution with γ = 1.5 and α = 8000 is 0.298.

FAILURE FUNCTION OF SYSTEM UNDER MULTIPLE FAILURE MECHANISMS

Let f1(t) and f2(t) be the probability density function of the system due to failure mechanism 1 and 2 respectively. Now the probability density function of the time-to-failure of the system caused by either of the failure mechanisms:

where, F1(t) and F2(t) the are failure function for failure mechanism 1 and 2 respectively. The failure function of the item under two different failure

mechanism is given by:

FAILURE FUNCTION, F(T):

RELIABILITY FUNCTION R(T):

HAZARD FUNCTION

MTTF OF DIFF DISTRIBUTION FUNCTIONS: