Normal distribution -Why $\Phi(a,b)=\Phi(b)-\Phi(a)$? I am reading "Probability" by Pitman and in the section that talks about normal distribution the book sais that $\Phi(a,b)=\Phi(b)-\Phi(a)$. Is this a definition or a theorem ? if it's a theorem why is it true ?

By definition, $\Phi(z)$ is the probability that $Z\le z$, where $Z$ is standard normal. And $\Phi(a,b)$ (for $a\le b$) is the probability that $a\le Z\le b$. But $$P(a\le Z\le b)=P(Z\le b)-P(Z
More informally, $\Phi(a)$ is the area under the standard normal curve, from $-\infty$ to $a$, and $\Phi(a,b)$ is the area under the curve, from $a$ to $b$. Now the area from $-\infty$ to $a$, plus the area from $a$ to $b$, is the area from $-\infty$ to $b$. In symbols, $$\Phi(a)+\Phi(a,b)=\Phi(b),$$ and therefore $\Phi(a,b)=\Phi(b)-\Phi(a)$.

The fact that $\Phi(a,b)=\Phi(b)-\Phi(a)$ is a theorem. However, it is _intuitively_ clear from the intended _meaning_ of $\Phi(a,b)$ and $\Phi(z)$.