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BUILT-IN MATHEMATICA SYMBOL
FailureDistribution
FailureDistribution[bexpr, {{x1, dist1}, {x2, dist2}, ...}]
represents the failure distribution for a system with events 
having reliability distribution 
where the top event occurs when the Boolean expression bexpr is True and event 
has occurred when 
is True.
Details
- FailureDistribution[bexpr, ...] corresponds to a fault tree specification.
- The Boolean expression bexpr is also known as the structure function for the system.
- Typical structure functions include:
-
Or gate 
And gate ![TemplateBox[{BooleanCountingFunction, paclet:ref/BooleanCountingFunction}, RefLink, BaseStyle -> InlineFormula][{k,n},n]](Files/FailureDistribution.en/8.png "TemplateBox[{BooleanCountingFunction, paclet:ref/BooleanCountingFunction}, RefLink, BaseStyle -> InlineFormula][{k,n},n]")
-out-of-
systemBooleanConsecutiveFunction[k, n] consecutive- 
-out-of-
system - The structure function bexpr can be any positive unate Boolean function.
- UnateQ[bexpr] can be used to test whether a Boolean expression is positive unate.
- The event reliability distributions
need to be univariate with PDF[disti, t] zero for t≤0. - For FailureDistribution[bexpr, ...] with event indicators
: -
xi=True indicates event 
has occurredxi=False indicates event 
has not occurred - The CDF at time t for FailureDistribution[bexpr, {{x1, dist1}, ...}] is given by Probability[bexpr/.{x1->t1≤t, ...}, {t1
dist1, ...}]. - FailureDistribution can be used with such functions as Mean, SurvivalFunction, HazardFunction, and RandomVariate.
Examplesopen all
Basic Examples (3)
Define a system whose top event occurs if either of the two events occur:
| In[1]:= |
Compute the survival function:
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Define a system whose top event occurs if both of the underlying events occur:
| In[1]:= |
Compute the survival function:
| In[2]:= |
| Out[2]= |  |
A structure with both an And and an Or gate:
| In[1]:= |
| In[2]:= |
| Out[2]= |  |
Mean and median time to failure:
| In[3]:= |
| Out[3]= |  |
| In[4]:= |
| Out[4]= |  |
The probability that the system fails before time 
:
| In[5]:= |
| Out[5]= |  |
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