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 ?
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 ?
Suppose $b\le a$. Then $ \Pr(X \le a) = \Pr(X \le b\text{ or }b
Subtract $\Pr(X\le b)$ from both sides of $(1)$ and it says: $ \Pr(X \le a) - \Pr(X\le b) = \Pr(b
If you replace $\Phi(a,b)$ and $\Phi(a),\Phi(a)$ with the integrals that define them, you get something like
$\int_a^b \circ~dx = \int_{-\infty}^b \circ~dx - \int_{-\infty}^a \circ~dx,$
which is elementary calculus.
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 The probability that $Z$ is exactly $a$ is $0$, so $P(Z, and the result follows.
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 $\Phi(z)$.