Pembahasan Ujian PAI: A70 – No. 14 – Mei 2018

• Lognormal Distribution
  \(E[X] = \exp \left( {\mu + \frac{{{\sigma ^2}}}{2}} \right)\)
• Rumus Umum
  \(E[X] = \frac{1}{5}\sum\limits_{t = 1}^5 {{x_t}} \)
  

\(\exp \left( {\mu + \frac{{{\sigma ^2}}}{2}} \right) = \frac{1}{5}\sum\limits_{t = 1}^5 {{x_t}} \)
\(\exp \left( {\mu + \frac{{{\sigma ^2}}}{2}} \right) = \frac{1}{5}\left( {500 + 1.000 + 1.500 + 2.500 + 4.500} \right)\)
\(\exp \left( {\mu + \frac{{{\sigma ^2}}}{2}} \right) = 2.000\)

• Lognormal Distribution
  \(E[{X^2}] = \exp \left( {2\mu + \frac{{{2^2}{\sigma ^2}}}{2}} \right)\)
  
• Rumus Umum
  \(E[{X^2}] = \frac{1}{5}\sum\limits_{t = 1}^5 {{x_t}^2} \)
  

\(\exp \left( {2\mu + \frac{{{2^2}{\sigma ^2}}}{2}} \right) = \frac{1}{5}\sum\limits_{t = 1}^5 {{x_t}^2} \)
\(\exp \left( {2\mu + \frac{{{2^2}{\sigma ^2}}}{2}} \right) = \frac{1}{5}\left( {{{500}^2} + {{1.000}^2} + {{1.500}^2} + {{2.500}^2} + {{4.500}^2}} \right)\)
\(\exp \left( {2\mu + \frac{{{2^2}{\sigma ^2}}}{2}} \right) = 6.000.000\)

\(\frac{{2\mu + \frac{{{2^2}{\sigma ^2}}}{2}}}{{\mu + \frac{{{\sigma ^2}}}{2}}} = \frac{{\ln 6.000.000}}{{\ln 2.000}}\)

\(\left( {2\mu + \frac{{{2^2}{\sigma ^2}}}{2}} \right)\ln 2.000 = \ln 6.000.000\left( {\mu + \frac{{{\sigma ^2}}}{2}} \right)\)
\(2\left( {\ln 2.000} \right)\mu + 2\left( {\ln 2.000} \right){\sigma ^2} = \left( {\ln 6.000.000} \right)\mu + \left( {\ln 6.000.000} \right)\frac{{{\sigma ^2}}}{2}\)
\(2\left( {\ln 2.000} \right){\sigma ^2} – \left( {\ln 6.000.000} \right)\frac{{{\sigma ^2}}}{2} = \left( {\ln 6.000.000} \right)\mu – 2\left( {\ln 2.000} \right)\mu \)
\(0,4054651081\mu = 7,398169905{\sigma ^2}\)

Solusi,

• \(\hat \mu = 7,398169905\)
  \(\hat \mu \cong 7,398\)
• \({\hat \sigma ^2} = 0,4054651081\)