I'm trying to figure out what different inequalities mean when testing the probability that a random variable is greater or less than a certain value. I've been searching around but I can't seem to find any questions that ask this specifically.
Say I have a CDF, $F(y)$, defined as $$F(y) = \left\{\begin{array}{ll} 0 & : y \lt 0\\ y & : 0 \le y \lt 1\\ 1 & : y \ge 1 \end{array} \right.$$ along with a random variable $X$, and some value $y$. I know that:
- $P(X \le y) = F(y)$
- $P(X \gt y) = 1 - F(y)$
- $P(y_1 \lt X \le y_2) = F(y_2) - F(y_1)$
But what method would I use to find the following?
- $P(X \lt y)$
- $P(X \ge y)$
- $P(y_1 \lt X \lt y_2)$
Does the strictness of the inequality have any effect in this context?
You can use either of the following:
$P(X < y) = P(X \le y) - P(X=y)$
$P(X < y) = \lim\limits_{z \uparrow y} P(X \le z)$ using the limit as $z$ approaches $y$ from below
If $X$ is a continuous random variable, i.e. the cumulative distribution function $F(y)$ is a continuous function, then $P(X=y)=0$ and $P(X < y) = P(X \le y) = F(y)$, so your second triple will have the same results as your first triple