Help with S&P probability 

*

*According to this model, what is the probability that the change in the S&P $500$ on a particular day is positive. We have $P(X>0)$ and then converting to standard normal table I get $P(Z>(0-0.75)/25=0.03$ because $\mu=0.75$ and the SD is $25$. Then using standard normal table we get $0.512$

*What is the third quartile quartile of daily changes in the S&P $500$? I know that $\mu$ and SD is the same, but would I just use $0.75$ as $X$? that would not make any sense because then you would just be dividing $0/25$? Also is there a way to do 1. without the standard table?

 A: Let me address your last question first: "Also is there a way to do (a) without the standard table?"
Well, one needs to know enough about the density of $X$ to determine the probability $\mathbb{P}(X > 0)$.  Where we get this information, is secondary.  A table is a convenient device for getting an approximation close enough.  Another way is to compute the appropriate integral of the density function, a task no less pleasant than looking into the table.
The third quartile of $X$ is such a value $q$ that $\mathbb{P}(X \leq q) \geq 0.75$.  Your normalized variable is
$$
Z = (X - \mu) / \mbox{SD},
$$
so the condition $X \leq q$, rewritten in terms of $Z$, would be
$$
Z \leq (q - \mu)/\mbox{SD}.
$$
Thus, you would be using the table to find the $q' = (q - \mu)/\mbox{SD}$ such that
$$
\mathbb{P}(Z \leq q') \geq 0.75.
$$
Once you've found the $q'$, solve for $q$.
A: Comment. Graph of $\mathsf{NORM}(\mu = 0.75, \sigma=25)$ showing 75th percentile $17.61224.$ (See initial comment moved just below from elsewhere on this page.)
1 - pnorm(0, .75, 25)
[1] 0.5119665    # area to right of vertical green line

qnorm(.75, .75, 25)
[1] 17.61224     # 75% of area is left of vertical red

May be confusing to see two .75s. Suppose you wanted the median (50th percentile). Should get mean $\mu = .75.$
qnorm(.5, .75, 23)
[1] 0.75


curve(dnorm(x,.75, 25), -75, 75, ylab="PDF", col="blue", lwd=2,
      main="NORM(0.75, 25")
 abline(v=17.61, col="red", lty="dotted", lwd=2)
 abline(h=0, col="green2");  abline(v=0, col="green2")

