21.4.7.6. Beta Distribution

The beta distribution is a useful distribution used when the upper and lower bounds of a random variable are defined as \(a\) and \(b\). Since the normal distribution is defined between \(-\propto\) and \(\propto\), the log-normal distribution is defined between 0 and \(\propto\), 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

\({{f}_{X}}\left( x \right)=\left\{ \begin{matrix} \frac{1}{B\left( q,r \right)}\frac{{{\left( x-a \right)}^{q-1}}{{\left( b-x \right)}^{r-1}}}{{{\left( b-a \right)}^{q+r-1}}}, & a\le x\le b \\ 0 & \text{elsewhere} \\ \end{matrix} \right.\)

where \(q\) and \(r\) are the parameters for distribution and \(B(q,r)\) is the beta function. The parameters \(q\) and \(r\) have the following relationship with Mean and Standard Deviation:

\({{\mu }_{X}}=a+\frac{q}{q+r}\left( b-a \right)\) and \(\sigma _{X}^{2}=\frac{qr}{{{\left( q+r \right)}^{2}}\left( q+r+1 \right)}{{\left( b-a \right)}^{2}}\)

Then, if we know mean, standard deviation and the distribution parameters \(q\) and \(r\), we can compute the lower and upper bounds as

\(a={{\mu }_{x}}-q\left( {{\sigma }_{x}}\sqrt{\frac{q+r+1}{qr}} \right)\) and \(b={{\mu }_{x}}+r\left( {{\sigma }_{x}}\sqrt{\frac{q+r+1}{qr}} \right)\)

When \(q=r=1\) , 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.