A new government introduces tuition of 10.000 Euro/year, increasing yearly with the inflation rate of 2%. At the same time, a prospective 'perpetual student' enrolls at the university. She expects that with 10% probability the government fails after 1 year, with 80% probability it is voted out of office after 4 years, and with 10% probability after 8 years (when she is still studying). In all these cases tuition will be cancelled immediately by the new government. What is the expectation value of the total amount of tuition she has to pay?
Hello
My discrete probability space $M:=\{1,4,8\}$
$P(t)$..total tuition after t year
Expected value $E(P_F)= 0.1*P(1)+0.8*P(4)+0.1*P(8)$
$P(0)=10.000$
$P(1)=10.00(1+0.02)+10.000$
$P(2)=10.00(1+0.02)*(1+0.02)+P(1)=10.00(1+0.02)^2+10.00(1+0.02)+10.000$
$P(3)=10.00(1+0.02)^3+P(2)=10.000(1.02)^3+10.00(1+0.02)^2+10.00(1+0.02)+10.000$
$P(4)=10.000*\sum_{j=0}^4 (1+0.02)^j=10.000* \frac{1- 1.02^5}{1-1.02} $
$E(P_F)= 0.1* 2.02*10.000+0.8*10.000* \frac{1- 1.02^5}{1-1.02} + 0.1*10.000* \frac{1- 1.02^9}{1-1.02} $
Ist this correct?