How to evaluate values of a function $$\Phi(x)=\frac{1}{\sqrt {2\pi}}\int_{-\infty}^xe^{-t^2/2}dt$$ using the following table:
I am given the example $\Phi(1.56)=0.9406$
This example is easy because we can directly read the value (two decimal places). In the $x$ column, we find $1.5$, and than in the $x$ row we find $6$.
What would be the value of $\Phi(1.563),\Phi(1.567),…$
Also, how to evaluate $\Phi$ function of transcendental numbers using the table?
What would be the value of $\Phi(\pi),\Phi(e),…$
Best Answer
The usual practice is to interpolate the table values. Here is an example of a linear interpolation to find $\Phi(0.7535)$. We look up the adjacent entries on either side to $0.7535$, namely $\Phi(0.75) = 0.7734$ and $\Phi(0.76) = 0.7764$. Then we note that $0.7535$ is $35\%$ of the way along the interval from $0.75$ to $0.76$, so $\Phi(0.7535)$ is also the same proportion between $\Phi(0.75)$ and $\Phi(0.76)$. More concretely, we have $$\frac{0.7535 - 0.75}{0.76 - 0.75} = 0.35,$$ hence linear interpolation gives the corresponding relationship $$\frac{\Phi(0.7535) - \Phi(0.75)}{\Phi(0.76) - \Phi(0.75)} = 0.35.$$ Solving for $\Phi(0.7535)$ we get $$\begin{align*} \Phi(0.7535) &\approx (0.35)(\Phi(0.76) - \Phi(0.75)) + \Phi(0.75) \\ &= (0.35)(0.7764 - 0.7734) + 0.7734 \\ &= 0.77445. \end{align*}$$ The same concept applies to other values. In general, if we have a continuous function $f(x)$ that is reasonably linear on some interval $[a,b]$, then for any $c \in [a,b]$, we have $$\frac{f(c) - f(a)}{f(b) - f(a)} \approx \frac{c-a}{b-a},$$ or $$f(c) \approx \frac{c-a}{b-a} (f(b) - f(a)) + f(a).$$