Var(X)=E[Var(X|Y)]+Var[E(X|Y)]=.75(1.41*10^12)+(2.3M-0)^2(.25)(.75)=2.049375*10^12=
(1.0575+.991875)*10^12=2.049375*10^12
standard deviation= sqrt(2.049375*10^12)=$1,431,564 (to the nearest dollar)
A company has a product liability lawsuit. It can settle out of court for 1.2 Million.
If it elects to go to court the jury has a 75% of finding the company liable.
If liable, the company will pay either 1 or 4 Million with probability .3 and 2 Million with probability .4.
The Expected cost of going to court is therefore .75[1M(.3)+2M(.4)+4M(.3)]+(.25*0)=1.725M
In order to compute the variance, I calculated the second moment as follows
.75[(1M)^2*.3+(2M)^2*.4+(4M)^2*.3)]=5.025*10^12
So the Variance of the cost of going to court would be 5.025*10^12-(1.725M)^2=2.049375*10^12, and the standard deviation would be $1,431,564. I'm
fairly certain this is correct, however, I am curious as to how the Variance would have been calculated using the second part of THE DOUBLE EXPECTATION THEOREM, that is,
VAR(X)=E[VAR(X|Y)]+VAR[E(X|Y)].