I am confused that which probability does the integral $\int_{0}^\infty \text{(pdf of gaussian) } dx$ equal to?
(1) the probability $P(X\geq 0)$
(2) $P(X>0)$
I believe it is equal to (1) , then my follow-up question is how to obtain the probability (2)?
This confused my a lot, since in my case, i would need to prove a variable is strictly positive, there is no chance for it to be zero.
The statement that $X$ is a $N(0,1)$ random variable implies, among other things, that $P[X\ge0]=P[X>0] = 1/2$ and $P[X=0]=0$.
But, in the standard measure-theoretic formulation of what a random variable is, one cannot tell if the event $[X=0]$ is the empty set, or just a set that happens to have probability $0$. One cannot conclude from $X\sim N(0,1)$ that $X=0$ is impossible, only that $P[X=0]=0$.
Suppose (in the notation of the cited Wikipedia page) that the underlying probability space $(\Omega,\mathcal F, P)$ is $((0,1),\mathcal B(0,1), \lambda)$, where $\mathcal B(0,0)$ is the Borel sets in $(0,1)$ and $\lambda$ is Lebesgue measure, and the random variable $X(\omega)=\Phi^{-1}(\omega)$, where $\Phi$ is cdf of the $N(0,1)$ distribution. Now, define a new random variable $Y$ by the formula $$Y(\omega)=\begin{cases}X(\omega)&\omega\ne1/2\\17&\omega=1/2.\end{cases}$$
The random variables are (referring to the cited Wikipedia paragraph) equal in distribution (both are $N(0,1)$-distributed), equal almost surely (because $P[X\ne Y]=0$), but not equal as functions on $\Omega$. The event $[X=0]$ is the set $\{1/2\}\subset \Omega$, which has Lebesgue measure $0$, but the event $[Y=0]$ is the empty set, which also has Lebesgue measure $0$. The event $[X=17]$ is the set $\{\Phi(17)\}$ which has Lebesgue measure $0$, and the event $[Y=17]$ is the set $\{1/2,\Phi(17)\}$ whose Lebesgue measure is also $0$.