The beta distribution is a useful distribution used when the
upper and lower bounds of a random variable are defined as 
and 
. Since the normal distribution is
defined between 
and
, the log-normal distribution
is defined between 0 and , it
is sometimes difficult to apply to the engineering random variable defined only
in the given upper and lower limit range. The probability density function of
the beta distribution is
where 
and 
are the parameters for distribution
and 
is the beta
function. The parameters and have the following relationship with
Mean and Standard Deviation:
and
Then, if we know mean, standard deviation and the
distribution parameters and , we can compute the lower and upper bounds
as
and
When 
, the beta distribution becomes the
uniform distribution.
Reference
1. Rao, S.S., Reliability-Based Design, McGraw-Hill, Inc., New 0York, 1992.
2. Haldar, A, and Mahadevan, S., Probability, Reliability, and Statistical methods in Engineering Design, John Wiley & Sons, Inc., 2000.