# JAM 2022 [ 41 -50]

If X be a random variable denoting the amount of loss in a business. The moment generating function of X is

M(t) = \left( \frac{2}{2-t} \right)^2 , t > 2

If an insurance policy pays 60% of the loss, then the variance of the amount paid by the insurance policy equals _________________ (round off 2 decimal places)

E(X) = M'(0) = \left[ 8 (2-t)^{-3} \right]_{t=0}=1 \\
E(X^2)= M''(0) =  \left[ 24 (2-t)^{-4} \right]_{t=0}=\frac{24}{16}=1.5 \\
V(X)=1.5-1=0.5

As per question, we need to find the variance of the amount paid by the insurance policy, which is 60% of the loss. Hence,

V(0.60X) = 0.36*0.5 = 0.18