Question

] Let X denote the size of a surgical claim and let Y denote the size of the associated hospital claim. An analyst is using a model in which Var[X] = 2.4, E[Y ] = 7, E[Y^2 ] = 51.4 and Var[X + Y ] = 8. If a 20% increase is added to the hospital portion of the claim, find the variance of the new total combined claim

Answer 1

We are given that 20% increase is added to the hospital portion of the claim, therefore the new Y here is given as:

Y2 = 1.2Y

We are given here that:
Var(X) = 2.4,
Var(Y) = E(Y2) - [E(Y)]2 = 51.4 - 72 = 2.4
Therefore, Var(Y2) = Var(1.2Y) = 1.22 Var(Y) = 1.44*2.4 = 3.456

We are also given here that:
Var(X + Y) = 8
Var(X) + Var(Y) + 2Cov(X, Y) = 8
2.4 + 2.4 + 2Cov(X, Y) = 8
Cov(X, Y) = (8 - 4.8) / 2 = 1.6

Therefore, Cov(X, Y2) = Cov(X, 1.2Y) = 1.2Cov(X, Y) = 1.2*1.6
= 1.92

Therefore, the new Variance here is computed as:
Var(X + Y2) = Var(X) + Var(Y2) + 2Cov(X, Y2)

= 2.4 + 3.456 + 2*1.92

= 9.696

Therefore 9.696 is the required variance here.