Random variable and maximum metric.

Question

Let $\Omega := [0,1] \times [0,1]$. Consider on $\sigma$-algebra Borel sets with Lebesgue measure .Let $X(w)$ describe the distance (in maximum metric) between the point and the nearest corner of the square. Is $X(w)$ a random variable? If so, find probability distribution and CDF. I need help,beacuse it is quite difficult.

Answer 1

Hint

Let $\omega =(\omega _1,\omega _2)\in \Omega $. Then,$$X(\omega )=\inf\Big\{\sup\{\omega _1,\omega _2\}, \sup\{\omega _1,1-\omega _2 \}, \sup\{1-\omega _1,1-\omega _2\}, \sup\{1-\omega _2,\omega_2\}\Big\}.$$

Added

Set \begin{align*} X_1(\omega )&=\sup\{\omega _1,\omega _2\}\\ X_2(\omega )&=\sup\{\omega _1,1-\omega _2\}\\ X_3(\omega )&=\sup\{1-\omega _1,\omega _2\}\\ X_4(\omega )&=\sup\{1-\omega _1,1-\omega _2\}. \end{align*}

$$\mathbb P\{X\leq x\}=\mathbb P\{X\boldsymbol 1_{[0,1/2]^2}\leq x\}+\mathbb P\{X\boldsymbol 1_{[0,1/2]\times [1/2,1]}\leq x\}+\mathbb P\{X\boldsymbol 1_{[1/2,1]\times [0,1/2]}\leq x\}+\mathbb P\{X\boldsymbol 1_{[1/2,1]^2}\leq x\}.$$ Now $$\mathbb P\{X\boldsymbol 1_{[0,1/2]^2}\leq x\}=\mathbb P\{\omega \in [0,1/2]^2\mid X_1\leq x\}=\mathbb P\{\omega\in [0,1/2]^2\mid \omega _1\leq x, \omega _2\leq x \}=\mathbb P\{\omega _1\in [0,1/2]\mid \omega _1\leq x\}\mathbb P\{\omega _2\in [0,1/2]\mid \omega _2\leq x\},$$ the last inequality come from independence of $\omega _1\mapsto \omega _1$ and $\omega _2\mapsto \omega _2$. Notice that that r.v. follow uniform law on $[0,1/2]$.

The rest goes the same.