# FailureDistribution

FailureDistribution[bexpr,{{x1,dist1},{x2,dist2},}]

represents the failure distribution for a system with events xi having reliability distribution disti where the top event occurs when the Boolean expression bexpr is True and event xi has occurred when xi 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 -out-of- system 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 disti need to be univariate with PDF[disti,t] zero for t0.
• For FailureDistribution[bexpr,] with event indicators xi:
•  xiTrue indicates event xi has occurred xiFalse indicates event xi has not occurred
• The CDF at time t for FailureDistribution[bexpr,{{x1,dist1},}] is given by Probability[bexpr/.{x1->t1t,},{t1dist1,}].
• FailureDistribution can be used with such functions as Mean, SurvivalFunction, HazardFunction, and RandomVariate.

# Examples

open allclose all

## Basic Examples(3)

Define a system whose top event occurs if either of the two events occurs:

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Compute the survival function:

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Define a system whose top event occurs if both of the underlying events occur:

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Compute the survival function:

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A structure with both an And and an Or gate:

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Distribution functions:

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Mean and median time to failure:

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The probability that the system fails before time :

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