Understanding Uniform Random Variables: Comparing $X$ and $Y = 1-X$

[Francobati]

Let $X\sim U(0,1)$ and define $Y=1-X$. What statement is TRUE?
(1): $F_{X}(u)\neq F_{Y}(u)$, for every $u\epsilon \left [ 0,1 \right ]$;
(2): $Y$ is not a rv;
(3): $E(X+Y)=2$;
(4): $Y\sim U(0,1)$;
(5): none of the remaining statements.

 

[Jameson]

Hi Francobati,

I am glad to find your interesting statistics questions! On MHB we always want you to give some kind of explanation of what you have tried. We are not a site that just "gives answers". So when you make a new thread it is best to always show what you have tried or what you know about the topic. This helps others see where you are stuck and how to best help.

So all of that said, what have you tried? What do you think about (3) for example?

 

[Francobati]

Hello. Many thanks. You are absolutely right. I am studying probability, I am trying to read the theory, but unfortunately for the exercises and the applications I need somebody routes, I addresses, because the practice is very different from the theory and is at the same time useful to better understand the theory.

 

[Siron]

A good strategy is to check each statement separately. If $X \sim U(0,1)$ then it's distribution function $F_X$ is given by:
$$F_X(u) = \left \{ \begin{array}{lll} 0, \quad u <0 \\ u, \quad 0\leq u \leq 1 \\ 1, \quad u > 1 \end{array} \right.$$
(1): use the fact that $F_X(u) = \mathbb{P}(X \leq u)$. Therefore, $F_Y(u) = \mathbb{P}(Y \leq u) = P(1-X \leq u) = \ldots$.
(2): you can make use of statement (1)
(3): since $Y = 1-X$ it follows that $X+Y = \ldots$?
(4): again, make use of statement (1)