Implementation of the second derivative of a normal probability distribution function in python

Question

IF : The PDF of the normal distribution is:

scipy.stats.norm.pdf(x, mu, sigma) Its first derivative with respect to x would be:

scipy.stats.norm.pdf(x, mu, sigma)*(mu - x)/sigma**2

What would be the second derivation?

Answer 1

You can apply the product rule

f(x)*g(x) = f(x)*g'(x) + f'(x)*g(x)

Where f(x) = pdf(x, mu, sigma), and g(x)=(mu-x)/sigma**2.

Then f'(x) = f(x) * g(x)

And g'(x) = -1/sigma**2

Putting all to gether you have the second derivative of the PDF as

def second_derivative(x, mu, sigma):
  g = (mu - x)**2/sigma**2;
  return scipy.stats.norm.pdf(x, mu, sigma)*(g**2 - 1/sigma**2)