Expectation and Conditional Expectation

The discussion on probability and conditional probability (previous day's blog) is continued to expectation and conditional expectation.

Let Z be a random variable denoting money spent by the state on the cardiac care of a citizen below 40 years.

E(Z) is the average money spent by the state per citizen below 40 years on its cardiac care.

E(Z)=P(Y1)E(Z|Y1)+P(Y2)E(Z|Y2)

=(Probability of a Heart attack before 40)(Average money spent by the state on caradiac care of a citizen with an incidence of heart attack before 40 years)+(Probability of no heart attack before 40 years)(Average money spent by the state on cardiac care of a citizen with no incidence of heart attack before 40)

so,

Expectation=Probability(condition1)* Conditional expectation+Probability(condition2)* Conditional expectation