Expected Value of Order Statistics with non IDD R.V.

Question

Let $$\begin{align} F_X(x) =&~ (0.03x^2-0.002x^3)\,\big[ 0 \le x \le 10\big]+\big[x>10\big] \\F_Y(y) =&~ y/10 \; \big[0\le y\le10\big]+\big[y>10\big] \\F_Z(z) =&~ (1-e^{{-z}/{5}})\,\big[z \ge 0\big] \end{align}$$

Find $\mathsf E\big(\max(X,Y,Z)\big)$

My Work:

let $W = \max(X,Y,Z)$

then $$F_W(x) = (0.03x^2-0.002x^3)(x/10)(1-e^{\frac{-x}{5}})$$

Then using the fact that $E(X) = A + \int_A^\infty\big(1-F_X(x)\big)\;\mathrm dx$ I should be able to find the answer

However, I have no clue what to plug in for the limits. When using this with IDD, it's clear, but here it isn't.

Thank you in advanced for any help.

Answer 1

Note that $F_W(x)$ is the product you wrote, when $0\le x\le 10$, and $1-e^{-x/5}$ when $x\gt 10$. Now integrate as usual, breaking up the integral into two parts, $1$ to $10$ and $10$ to $\infty$.