Question

Claim amounts for wind damage to insured homes are independent random variables with common density f(x) = ( 3 x4 , x > 1 0 , otherwise where x is amount of claim in thousands. (a) find the probability that a claim is below average? [19/27] (b) suppose 3 claims will be made. what is the expected value of the largest of the three claims? [2.025] (c) suppose 3 claims will be made. what is the expected value of smallest of the three claims?[1.125]

Answer 1

$f_X(x)=\begin{cases}\dfrac3{x^4}&\text{for }x>1\\\\0&\text{otherwise}\end{cases}$

This distribution has expectation

$E[X]=\displaystyle\int_{-\infty}^\infty xf_X(x)\,\mathrm dx=\int_1^\infty\frac3{x^3}\,\mathrm dx=\frac32$

a. The probability that $X$ falls below the average/expectation is

$P\left(X$

b. Denote by $X_{(3)}$ the largest of the three claims $X_1,X_2,X_3$. Then the density of this maximum order statistic is

$f_{X_{(3)}}(x)=3f_X(x)F_X(x)^2$

where $F_X(x)=P(X\le x)$ is the distribution function for $X$. This is given by

$F_X(x)=\displaystyle\int_{-\infty}^xf_X(t)\,\mathrm dt=\begin{cases}0&\text{for }x$

So we have

$f_{X_{(3)}}(x)=\begin{cases}0&\text{for }x1\end{cases}$

and the expectation is

$E[X_{(3)}]=\displaystyle\int_{-\infty}^\infty xf_{X_{(3)}}(x)\,\mathrm dx=\int_1^\infty\frac9{x^3}\left(1-\frac1{x^3}\right)^2\,\mathrm dx=\frac{81}{40}=\boxed{2.025}$

c. Denote by $X_{(1)}$ the smallest of the three claims. $X_{(1)}$ has density

$f_{X_{(1)}}(x)=3f_X(x)(1-F_X(x))^2=\begin{cases}0&\text{for }x1\end{cases}$

so the expectation is

$E[X_{(1)}]=\displaystyle\int_{-\infty}^\infty xf_{X_{(1)}}(x)\,\mathrm dx=\int_1^\infty\frac9{x^9}\,\mathrm dx=\frac98=\boxed{1.125}$