Template:Lambda-beta parameter relationship

Lambda - Beta Parameter Relationship

Under the Crow-AMSAA (NHPP) model, the time to first failure is a Weibull random variable. The MTTF of a Weibull distributed random variable with parameters [math]\displaystyle{ \beta }[/math] and [math]\displaystyle{ \eta }[/math] is:

[math]\displaystyle{ MTTF=\eta \cdot \Gamma \left( 1+\frac{1}{\beta } \right) }[/math]

The parameter lambda is defined as:

[math]\displaystyle{ \lambda =\frac{1}{{{\eta }^{\beta }}} }[/math]

Using Eqn. (lambda eta relationship), the MTTF relationship shown in Eqn. (Weibull MTTF) becomes:

[math]\displaystyle{ MTB{{F}_{B}}=\frac{\Gamma \left( 1+\tfrac{1}{\beta } \right)}{{{\lambda }^{\left( \tfrac{1}{\beta } \right)}}} }[/math]

Or, in terms of failure intensity:

[math]\displaystyle{ {{\lambda }_{B}}=\frac{{{\lambda }^{\left( \tfrac{1}{\beta } \right)}}}{\Gamma \left( 1+\tfrac{1}{\beta } \right)} }[/math]