Mathematics: Claim amounts for wind damage to insured homes are independent random variables with common density function f(x) = 3x −4 for x 1, and f(x) = 0 otherwise, where x is the amount of a claim in thousands. Suppose 3 such claims are made. What is the expected value of the largest of the three claims?

it is 1 because 100 divided by itself is 1. step-by-step explanation:...

Claim amounts for wind damage to insured homes

jesser9

22.01.2022

The expected value of the largest of the three claims is $ 2025.00

Step-by-step explanation:

To solve the question, if we have the three claims given by

C₁, C₂, C₃ and the let the maximum claim be denoted by M = max(C₁, C₂, C₃)

If the expected value of M is expressed as E(M) and the common density function is F(m) with p,d,f f(m) = F'(m) of M

Therefore, F(m) = P (max C₁, C₂, C₃) ≤ M) = P (C₁ ≤m) × P (C₂ ≤m) × P (C₃ ≤m)

$(\int\limits^y_1 {3x^{-4}} \, dx )^3$ = $(1-\frac{1}{m^3} )^{3$

From which

F'(m) = $\frac{3}{m^4}\times 3\times (1-\frac{1}{m^3} )^{2$ = $\frac{9}{m^4} (1-\frac{1}{m^3} )^{2$

E(M) = $\int\limits^\infty_1 {m f(m)} \, dm = \int\limits^\infty_1 {\frac{9}{m^3} (1-\frac{1}{m^3} )^{2\, dm$

Which gives $9\times \int\limits^\infty_1 {\frac{1}{m^3} (1-\frac{1}{m^3} )^{2\, dm$

= $9\times \int\limits^\infty_1 { (\frac{1}{m^3} -\frac{2}{m^6} + \frac{1}{m^9} )\, dm$

= 9× $(\frac{1}{2}-\frac{2}{5}+\frac{1}{8})$ = $ 2.025 × 10³

=$ 2025.00

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