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Analytical and Computational Approach

This paper focuses on an analytical and computational approach to problem-solving in various industries.

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    Analytical and Computational Approach 3. fx(x) = (theta^2 + theta) x^(theta - 1) (1 - x) ; 0<x<1; theta > 0 a. Consider, 𝐸 ( 𝑋 ) = ∫ π‘₯ 𝑓 ( π‘₯ ) 𝑑π‘₯ 1 0 = ∫ π‘₯ ( 𝑑 β„Ž 𝑒𝑑 π‘Ž 2 + 𝑑 β„Ž π‘’π‘‘π‘Ž ) π‘₯ 𝑑 β„Ž π‘’π‘‘π‘Ž βˆ’ 1 ( 1 βˆ’ π‘₯ ) 𝑑π‘₯ 1 0 = ( 𝑑 β„Ž 𝑒𝑑 π‘Ž 2 + 𝑑 β„Ž π‘’π‘‘π‘Ž ) ∫ π‘₯ 𝑑 β„Ž π‘’π‘‘π‘Ž ( 1 βˆ’ π‘₯ ) 𝑑π‘₯ 1 0 = ( 𝑑 β„Ž 𝑒𝑑 π‘Ž 2 + 𝑑 β„Ž π‘’π‘‘π‘Ž ) ∫ π‘₯ 𝑑 β„Ž π‘’π‘‘π‘Ž βˆ’ π‘₯ 𝑑 β„Ž π‘’π‘‘π‘Ž + 1 𝑑π‘₯ 1 0 = ( 𝑑 β„Ž 𝑒𝑑 π‘Ž 2 + 𝑑 β„Ž π‘’π‘‘π‘Ž ) ( π‘₯ 𝑑 β„Ž π‘’π‘‘π‘Ž + 1 𝑑 β„Ž π‘’π‘‘π‘Ž + 1 βˆ’ π‘₯ 𝑑 β„Ž π‘’π‘‘π‘Ž + 2 𝑑 β„Ž π‘’π‘‘π‘Ž + 2 ) 0 1 = ( 𝑑 β„Ž 𝑒𝑑 π‘Ž 2 + 𝑑 β„Ž π‘’π‘‘π‘Ž ) βˆ— ( 1 𝑑 β„Ž π‘’π‘‘π‘Ž + 1 βˆ’ 1 𝑑 β„Ž π‘’π‘‘π‘Ž + 2 ) = 𝑑 β„Ž π‘’π‘‘π‘Ž βˆ— ( 1 + 𝑑 β„Ž π‘’π‘‘π‘Ž ) βˆ— ( 1 ( 𝑑 β„Ž π‘’π‘‘π‘Ž + 1 ) βˆ— ( 𝑑 β„Ž π‘’π‘‘π‘Ž + 2 ) ) = 𝑑 β„Ž π‘’π‘‘π‘Ž 𝑑 β„Ž π‘’π‘‘π‘Ž + 2 thus, π‘‹π‘π‘Žπ‘Ÿ βˆ— ( 𝑑 β„Ž π‘’π‘‘π‘Ž + 2 ) = 𝑑 β„Ž π‘’π‘‘π‘Ž π‘‹π‘π‘Žπ‘Ÿ βˆ— 𝑑 β„Ž π‘’π‘‘π‘Ž + ( 2 βˆ— π‘‹π‘π‘Žπ‘Ÿ ) βˆ’ 𝑇 β„Ž π‘’π‘‘π‘Ž = 0 𝑑 β„Ž π‘’π‘‘π‘Ž ( π‘‹π‘π‘Žπ‘Ÿ βˆ’ 1 ) = βˆ’ 2 π‘‹π‘π‘Žπ‘Ÿ 𝑑 β„Ž π‘’π‘‘π‘Ž = βˆ’ 2 βˆ— π‘‹π‘π‘Žπ‘Ÿ ( π‘‹π‘π‘Žπ‘Ÿ βˆ’ 1 )

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