I have been trying to write the code for the Cumulative Distribution function (CDF) of the normal distribution in C++. Since the cdf does not have a closed form solution of the integral, I was wondering what approximations are generally used for the implementation in any programming language. Can anyone please explain how the approximating function is equal to the integral/cdf in some detail? Thanks in advance.
[Math] Numerical Approximations to the Cumulative Distribution Function of the Normal Distribution
normal distribution
Related Solutions
Let $X$ and $Y$ be the standard normal random variables. Then $$ \mathbb{E}(\Phi(a X + b)) = \mathbb{E}( \mathbb{P}( Y \le a x + b \vert X = x ) ) = \mathbb{P}(Y- a X \le b ) $$ But the combination $Z = Y-a X$ also follows normal distribution (being a linear combination of normals), with zero mean and variance $\mathbb{E}((Y-a X)^2) = 1 + a^2$. Hence $$ \mathbb{E}(\Phi(a X + b)) = \Phi\left(\frac{b}{\sqrt{1+a^2}}\right) $$
Here is numerical checks:
In[14]:= With[{a = 3.,
b = 1/2}, {NExpectation[CDF[NormalDistribution[], a x + b],
x \[Distributed] NormalDistribution[]],
CDF[NormalDistribution[], b/Sqrt[1 + a^2]]}]
Out[14]= {0.562816, 0.562816}
Here is a quick example in c++ of an implementation. Let me know if it helps. If not I will remove the answer :).
Best Answer
Abramowitz and Stegun give a number of approximations. One with only five constants is accurate to $1.5 \cdot 10^{-7}$ Numerical Recipes page 221 has an expansion that quotes slightly better errors. Neither explains where they come from, but you could look at the references.