I am doing joint probability and i came across a problem with bounds of the joint probability.
Let $X, Y, Z$ be independent and uniformly distributed over $(0, 1)$. Compute $Pr(X ≥ Y Z)$.
From the solutions:
\begin{align*} Pr\{X \geq YZ\} &= \int \int \int_{x \geq yz} f_{XYZ}(x, y, z) \, dx \, dy \, dz \\ &= \int_0^1 \int_0^1 \int_{yz}^1 1 \, dx \, dy \, dz \\ &= \int_0^1 \int_0^1 (1 - yz) \, dy \, dz \\ &= \int_0^1 \left(1 - \frac{z}{2}\right) \, dz \\ &= \frac{3}{4}. \end{align*}
What i am unsure of is the step: \begin{align*} &= \int \int \int_{x \geq yz} f_{XYZ}(x, y, z) \, dx \, dy \, dz \\ &= \int_0^1 \int_0^1 \int_{yz}^1 1 \, dx \, dy \, dz \\ \end{align*}
Why is intuitive reasoning on why the first 2 integral bounds are from 0 to 1. I am thinking of it as they are uniformly distributed hence defined over 0 to 1 as stated, but I dont feel convinced by it.
Thank you