An insurance policy pays a total medical benefit consisting of two parts for each claim. Let X represent the part of the benefit that is paid to the surgeon, and let Y represent the part that is paid to the hospital. The variance of X is 5000, the variance of Y is 10,000, and the variance of the total benefit, X + Y, is 17,000. Due to increasing medical costs, the company that issues the policy decides to increase X by a flat amount of 100 per claim and to increase Y by 10% per claim. Calculate the variance of the total benefit after these revisions have been made

= 19300

Step-by-step explanation:

Each claim consists of two parts = X + Y

where

X = the benefit that is paid to the surgeon and

Y = benefit that is paid to the hospital

V (X) = 5000, V (Y) = 10000 and V (X+Y) = 17000

So V (X+Y) = V (X) + V (Y) + 2cov (X, Y)

17000 = 5000 + 10000 + 2 cov (X, Y)

17000 - 15000 = 2cov (X, Y)

2000 = 2cov (X, Y)

cov (X, Y) = 1000

Now X is increased by flat Rs. 100 per claim and Y by 10% per claim

total benefit = X+100+Y+0.1Y = X+100 + 1.1Y

V (total benefit) = V (X) + 1.1²V (Y) + 2 (1.1) cov (X, Y) [ V (aX+bY)

= a²V (X) + b²V (Y) + 2abcov (X, Y) and V (X+c) = V (X) ]

= 5000 + (1.21*10000) + (2.2*1000)

= 5000 + 12100 + 2200

= 19300