(Answered)-Let X be a discrete random variable and Y an independent - (2025 Updated Original AI-Free Solution

Question

1. Let?X?be?a?discrete?random?variable?and?Y?an?independent?Uniform[0,?1]?random?variable.?Show?that?the?random?variable?Z?=?{X?+?Y?}?is?also?uniform?and?it?is?independent?of?X.?
2. Recall?that?{a}???[0,?1)?is?the?fractional?part?of?a.???????{a}?=?a????a??where??a??is?the?largest?integer?not?bigger?than?a.?