Correct Answer - Option 3 :
\(\frac{1}{4}\)
Concept:
For uniform distribution:
\({\rm{f}}\left( {\rm{x}} \right) = \frac{1}{{{\rm{b}} - {\rm{a}}}}\) for a < x < b
\({\rm{E}}\left( {{{\rm{X}}^n}} \right) = \mathop \smallint \limits_{0}^1 {{\rm{X}}^n}{\rm{f}}\left( {\rm{X}} \right){\rm{dX}}\)
Calculation:
Given:
a = 0, b = 1,
\({\rm{f}}\left( {\rm{x}} \right) = \frac{1}{{{\rm{b}} - {\rm{a}}}}\)
\({\rm{f}}\left( {\rm{x}} \right) = \frac{1}{{1 - 0}} = 1\)
\(\begin{array}{l} {\rm{E}}\left( {{{\rm{X}}^3}} \right) = \mathop \smallint \limits_{0}^1 {{\rm{X}}^3}{\rm{f}}\left( {\rm{X}} \right){\rm{dX}}\\ = \left[ {\frac{{{{\rm{X}}^4}}}{4}} \right]_{0}^1 = \frac{1}{4} \end{array}\)
