2) Consider rolling one fair six-sided die, so that $S = \{1, 2, 3, 4, 5, 6\}$, and $P(s) = 1/6$ for all $s \in S$. Let $X$ be the number showing on the die, so that $X(s) = s$ for $s \in S$. Let $Y = X^2$. Compute the cumulative distribution function $F_Y (y) = P(Y \leq y)$, for all $y \in R^1$. Verify explicitly that properties (a) through (d) of Theorem 2.5.2 are satisfied by this function $F_Y$ .
This is theorem 2.5.2
Theorem 2.5.2 Let $F_X$ be the cumulative distribution function of a random variable $X$. Then
(a) $0 \leq F_X(x) \leq 1, \forall x$
(b) $F_X (x) ≤ F_X (y)$ whenever $x ≤ y$ (i.e., $F_X$ is increasing)
(c) $\lim_{x\to+\infty} F_X(x) = 1$
(d) $\lim_{x\to-\infty} F_X(x) = 0$
SOLUTION:
$$F_X(x) = $$
\begin{cases} 0 & x < 1 \\ 1/6 & 1\leq x < 4 \\ 2/6 & 4 \leq x < 9 \\ 3/6 & 9 \leq x < 16 \\ 4/6 & 16 \leq x < 25 \\ 5/6 & 25 \leq x < 36 \\ 1 & 36 \leq x \end{cases}
Properties (a) and (b) follow by inspection. Properties (c) and (d) follow since $F_y(y) = 0$ for $y < 1$, and $F_y(y) = 1$ for $y > 36$
Could someone please explain this solution because I don't understand how they got $F_X(x)$.
Like for example, the first one is if $F(x < 1) = 0$. What is $x$?
This is a piecewise function, which has the form $g(x) =\begin{cases} a &:& x\in A\\ b &:& x\in B \\ &\vdots&\\ c &:& \text{otherwise}\end{cases}$
The domain of function $g$ is partitioned into disjoint sets $A,B,$ et cetera, and the value held by $g$ varies according to which part contains the argument.
It means that when $x\in A$, then $g(x)$ has value of $a$, else when $x\in B$, then $g(x)$ has value $b$, and so forth, with $g(x)$ having value $c$ when no other part holds.
So when $x<1$ the cummulative distribution function has value $0$.
Remark: the solution is wrong, in that it should be $F_Y(y)$ rather than $F_X(x)$; a typographic error. Just replace the letters to match the question.
$$F_Y(y) = \begin{cases} 0 &:& \qquad y < 1 \\ 1/6 &:& 1\leq y < 4 \\ 2/6 &:& 4 \leq y < 9 \\ 3/6 &:& 9 \leq y < 16 \\ 4/6 &:& 16 \leq y < 25 \\ 5/6 &:& 25 \leq y < 36 \\ 1 &:& 36 \leq y \end{cases}$$