The cummulative distribution function or cdf of a random variable $X$, denoted by $F_X(x),$ is defined by \[F_X(x) = P_X(X\leq x), \; \text{ for all }x.\]
Theorem 1.7.1
The function $F(x)$ is a cdf if and only if the following three conditions hold:
a.) $\lim_{x\rightarrow -\infty} F(x) = 0$ and $\lim_{x\rightarrow \infty} F(x) = 1.$
b.) $F(x)$ is a nondecreasing function of $x$.
c.) $F(x)$ is right continuous; that is, for every number $x_0, \lim_{x\rightarrow x_0^+} F(x) = F(x_0).$
a.) $\lim_{x\rightarrow -\infty} F(x) = 0$ and $\lim_{x\rightarrow \infty} F(x) = 1.$
b.) $F(x)$ is a nondecreasing function of $x$.
c.) $F(x)$ is right continuous; that is, for every number $x_0, \lim_{x\rightarrow x_0^+} F(x) = F(x_0).$
Recall: The partial sum of the geometric series is \[\sum_{k=1}^n t^{k-1} = \frac{1-t^n}{1-t}, \; t\neq 1.\]
A random variable $X$ is continuous if $F_X(x)$ is a continuous function of $x.$ A random variable $X$ is discrete if $F_X(x)$ is a step function of $x.$
The random variables $X$ and $Y$ are identically distributed if, for every set $A\in \mathcal{B^1},P(X\in A) = P(Y \in A).$
Note: Two random variables that are identically distributed are not necessarily equal. I.e., identically distributed does not imply $X=Y.$
Theorem 1.7.2
The following two statements are equivalent:
a.) The random variables $X$ and $Y$ are identically distributed.
b.) $F_X(x) = F_Y(x)$ for every $x.$
a.) The random variables $X$ and $Y$ are identically distributed.
b.) $F_X(x) = F_Y(x)$ for every $x.$