91Ó°ÊÓ

X be a random variable that follows a beta distribution with parameters \(\alpha \) and \(\beta \)

Pdf of beta distribution with parameters\(\alpha ,\beta > 0\)is:

\(f\left( {x|\alpha ,\beta } \right) = \left\{ \begin{array}{l}\frac{{\left| \!{\overline {\, {\left( {\alpha + \beta } \right)} \,}} \right. }}{{\left| \!{\overline {\, {\left( \alpha \right)} \,}} \right. \left| \!{\overline {\, {\left( \beta \right)} \,}} \right. }}{x^{\alpha - 1}}{\left( {1 - x} \right)^{\beta - 1}}\;\;\;\;\;for\,0 < x < 1\\0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Otherwise\end{array} \right.\)

Therefore

\({f'}\left( {x|\alpha ,\beta } \right) = \frac{{\left| \!{\overline {\, {\left( {\alpha + \beta } \right)} \,}} \right. }}{{\left| \!{\overline {\, {\left( \alpha \right)} \,}} \right. \left| \!{\overline {\, {\left( \beta \right)} \,}} \right. }}\left[ {\left( {\alpha - 1} \right)\left( {1 - x} \right) - \left( {\beta - 1} \right)x} \right]{x{\alpha - 2}}{\left( {1 - x} \right){\beta - 2}}\)

Hence

\({f'}\left( {x|\alpha ,\beta } \right) = 0\) and

\(x = \frac{{\left( {\alpha - 1} \right)}}{{\left( {\alpha + \beta - 2} \right)}}\)

It can be verified that if\(\alpha > 1\)and\(\beta > 1\), then\(f\left( {x|\alpha ,\beta } \right)\)is actually a maximum for this value of x.

Hence the mode of the beta distribution is \(\frac{{\left( {\alpha - 1} \right)}}{{\left( {\alpha + \beta - 2} \right)}}\).