Pembahasan Ujian PAI: A70 – No. 24 – November 2016

Hitunglah estimasi Buhlmann credibility dari observasi kedua, jika diketahui observasi pertama adalah 1

• \(Z = \beta = \frac{{Cov\left( {X,Y} \right)}}{{Var\left( X \right)}}\)
• \({P_C} = \alpha + Z\bar X\)
• \(\alpha = \left( {1 – Z} \right)E\left[ X \right]\)

• \(E\left[ X \right] = \frac{{1 + 2 + 3}}{3} = 2\)
• \(Var\left( X \right) = \frac{{{{\left( {1 – 2} \right)}^2} + {{\left( {2 – 2} \right)}^2} + {{\left( {3 – 2} \right)}^2}}}{3} = \frac{2}{3}\)
• \(Cov\left( {X,Y} \right) = \frac{{\left( {1 – 2} \right)\left( {1.5 – 2} \right) + \left( {2 – 2} \right)\left( {1.5 – 2} \right) + \left( {3 – 2} \right)\left( {3 – 2} \right)}}{3} = \frac{1}{2}\)

• \(Z = \beta = \frac{{Cov\left( {X,Y} \right)}}{{Var\left( X \right)}} = \frac{{\frac{1}{2}}}{{\frac{2}{3}}} = 0.75\)
• \(\alpha = \left( {1 – 0.75} \right)\left( 2 \right) = 0.5\)
• \({P_C} = \alpha + Z\bar X = 0.5 + \left( {0.75} \right)\left( 1 \right) = 1.25\)