# How do the following r.v. notations read in plain English?

Source: Schaum's Outline of Theory and Problems of Probability, Random Variables, and Random Processes. Page-122

How do the following notations read in plain English?

• $$(Y\le y)=[g(X)\le y]=(X \in D_Y)$$
• $$F_Y(y)=P(Y\le y)=P[g(X) \le y] = P(X \in D_Y)$$
• $$\displaystyle F_Y(y)=\int_{D_Y} f_X(x)\, dx$$
• For $F_Y$, see Cumulative distribution function associated to the real-valued random variable $Y$. – Mauro ALLEGRANZA Sep 26 '18 at 9:24
• $F_Y(y) = \text {Pr}(Y \le y)$ is the probability that the random variable $Y$ takes on a value less than or equal to $y$. A random variable is defined as a function that maps the outcomes of unpredictable processes to numerical quantities (labels), typically real numbers. Thus, $(Y \le y)$ is an outcome or event. – Mauro ALLEGRANZA Sep 26 '18 at 9:25
• @MauroALLEGRANZA, what is the deal with square brackets? – user366312 Sep 26 '18 at 15:04
• What do you mean ? How to "read in plain English" square brackets ? – Mauro ALLEGRANZA Sep 26 '18 at 15:05
• @MauroALLEGRANZA, What do you mean? --- I meant: why is it written as $P[g(X) \le y]$? Why not $P(g(X) \le y)$? – user366312 Sep 26 '18 at 15:10