If $Y = 1 - X$ then is the cdf of $Y$, 1 for y ≥ 1?

Question

$$F_{X}(x)=\begin{cases} 0, & x<0 \\ x, & 0 \le x < 1 \\ 1, & x ≥ 1. \end{cases}$$

The question is contained in a text book:

This is how I have proceeded: $P(y≥1) = P[(1-x)≥1] = P(x\leq 0) = 0$.

I don't know how to get $1$ .

Answer 1

For the cumulative distribution function of $Y$, i.e. $F_Y(y)=P(Y \le y)$ when $y=1$, you should be aiming for

$$F_Y(1)=P(Y\le 1) = P[(1-X)\le 1] = P(X\geq 0) = 1$$

More generally you get $F_Y(y)=P(Y\le y) = P[(1-X)\le y] = P(X\geq 1-y)$, which is also $1$ when $y \ge 1$