Lognormal Distribution
The two parameters required for the lognormal distribution are the mean and standard deviation. The lognormal distribution is similar to the normal distribution, except that the logarithm of the values of random variables, rather than the values themselves, are assumed to be normally distributed. Thus, all values are positive, and the distribution is skewed to the left.
The lognormal distribution is probably the most significant competitor to the Weibull distribution. It is frequently used in engineering for metal fatigue testing, maintainability data (time to repair), chemical process equipment failures and repairs, some material characteristics, and nonlinear, accelerating deterioration. When the time to failure results from the multiplication of effects, the lognormal distribution is recommended. For example, in the case of progressive deterioration, a crack forms due to stress, and the stress increases as the crack grows. Outside of-engineering, the lognormal distribution is often used to analyze financial information, such as personal incomes, inheritances, and bank deposits.
Calculations
The probability density function, f(t), and the survival function, that is reliability, R(t) ≡ 1 − F(t), with respect to time t, follows for computing lognormal distributions.
Where:
t .≥ 0, ∞ < μ < ∞, and σ > 0
In this case, μ and σ are the mean and standard deviation of the natural logarithm of the time to failure (lognormal random variable).
For the Weibull module, the input variables are log mean antilog (MuAL) and standard deviation factor (SigF).
MuAL = exp(μ); MuA (L > 0)
SigF = exp{σ}; Sig(F > 1)
If a location parameter exists, see Location Parameter for additional calculation information.
For the RBD module, the inputs are Mu and Sigma for a block with a lognormal failure or repair distribution.