Pembahasan Ujian PAI: A70 – No. 4 – November 2016

Banyaknya klaim dan besarnya klaim saling bebas (independent) pada setiap jenis pertanggungan

• \(Z = \frac{{na}}{{na + v}}\)
• \(\mu = E\left[ {\mu \left( \Theta \right)} \right] = E\left[ {E\left( {\left. {{X_j}} \right|\Theta } \right)} \right]\) ekspektasi dari hypothetical mean
• \(a = Var\left[ {\mu \left( \Theta \right)} \right] = Var\left[ {E\left( {\left. {{X_j}} \right|\Theta } \right)} \right]\) varians dari hypothetical mean
• \(v = E\left[ {v\left( \Theta \right)} \right] = E\left[ {Var\left( {\left. {{X_j}} \right|\Theta } \right)} \right]\)ekspektasi dari process varians
• Agregat:
  \(E\left[ S \right] = E\left[ N \right]E\left[ X \right]\) \(Var\left[ S \right] = E\left[ N \right]Var\left[ X \right] + Var\left[ N \right]E{\left[ X \right]^2}\)
• Pareto:
  \(E\left[ X \right] = \frac{\theta }{{\alpha – 1}}\) \(Var\left( X \right) = \frac{{2{\theta ^2}}}{{\left( {\alpha – 1} \right)\left( {\alpha – 2} \right)}} – {\left( {\frac{\theta }{{\alpha – 1}}} \right)^2}\)

• \(E\left[ N \right] = 0.1\) dan \(Var\left[ N \right] = \left( {0.1} \right)\left( {0.9} \right) = 0.09\)
• \(E\left[ X \right] = \frac{{50}}{{3 – 1}} = 25\) dan \(Var\left( X \right) = \frac{{2\left( {{{50}^2}} \right)}}{{\left( 2 \right)\left( 1 \right)}} – {25^2} = 1875\)

Pada tertanggung B

• \(E\left[ N \right] = 0.2\) dan \(Var\left[ N \right] = \left( {0.2} \right)\left( {0.8} \right) = 0.16\)
• \(E\left[ X \right] = \frac{{60}}{{3 – 1}} = 30\) dan \(Var\left( X \right) = \frac{{2\left( {{{60}^2}} \right)}}{{\left( 2 \right)\left( 1 \right)}} – {30^2} = 2700\)