Height and Area of the Bell Curve

A mill fills bags of flour, and the fill weight $X$ (in grams) is modelled as normal: $X \sim N(\mu, \sigma^2)$ . For any weight $x$ , quality control asks two different questions.

How dense is the curve at $x$ ? That is the PDF value $f(x)$ , the height of the bell curve at $x$ :

f(x) = \frac{1}{\sigma \sqrt{2\pi}}\, e^{-\frac{(x-\mu)^2}{2\sigma^2}}

What fraction of bags weigh at most $x$ ? That is the CDF value $F(x) = P(X \le x)$ , the area under the curve to the left of $x$ .

The normal CDF has no formula built only from arithmetic and powers. Python's standard library does ship the error function math.erf, though, and the CDF of the standard normal $Z \sim N(0, 1)$ can be written with it:

\Phi(z) = \frac{1}{2}\left[1 + \operatorname{erf}\!\left(\frac{z}{\sqrt{2}}\right)\right]

Task: write normal_pdf_cdf(x, mu, sigma) that returns the tuple (f(x), F(x)) for $X \sim N(\mu, \sigma^2)$ , each rounded to 4 decimal places.

sigma is the standard deviation $\sigma$ (always greater than $0$ ), not the variance.
$\Phi$ above is the CDF of $N(0, 1)$ only; your function must work for any $\mu$ and $\sigma$ .
Use only the standard library.

For example, a mill aiming at $\mu = 512$ g with $\sigma = 15$ g wants to know how often a bag comes in at or under the $500$ g printed on the label: that is the second number returned by normal_pdf_cdf(500, 512, 15).

Height and Area of the Bell Curve

Examples

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