Respuesta :
The moment-generating functions for the loss distributions of the cities are
MJ (t) = (1 − 2t)−3, MK(t) = (1 − 2t)−2.5, ML(t) = (1 − 2t) −4.5
Let X represent the combined losses from the three cities. Calculate E(X3)
Solution:
Let J, K, L denote the losses from the three cities. Then X = J + K + L. Since J, K,
L are independent, the moment-generating function for their sum, X, is equal to the product of the individual moment-generating functions, i.e.,
MX(t) = MK(t)MJ (t)ML(t) = (1 − 2t)-3-2.5-4.5=(1-2t)-10
Differentiating this function, we get
M'(t) = (−2)(−10)(1 − 2t)−11
M''(t) = (−2)2(−10)(−11)(1 − 2t)−12
M'''(t) = (−2)3(−10)(−11)(−12)(1 − 2t)−13
Hence, E(X3) = M m/x(0) = (−2)3(−10)(−11)(−12) = 10, 560.
Therefore the answer is the moment-generating functions for the loss distributions of the cities are 10, 560.
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