Expectation of minimum in piecewise density

Let $(X, Y, Z)$ be a random vector having the joint probability density function $$f(x, y, z)=\begin{cases} \frac{1}{2xy}, & \text{if }0 < z < y < x < 1, \ \frac{1}{2x^2}, & \text{if }0 < z < x < y < 2x < 2, \ 0, & \text{otherwise.} \end{cases}$$ Then, which one of the following statements is FALSE?