Normal Distribution PDF

Implement the probability density of the normal distribution.

f(x) = 1 / (sigma * sqrt(2 * pi)) * exp(-(x - mu)^2 / (2 * sigma^2))

x may be a scalar or an array; return the same shape.

This density is the reason squared error is the loss for regression. Maximising the log likelihood of Gaussian noise is minimising (x - mu)^2, and everything else in the formula is a constant that the gradient throws away.

Function signature

def normal_pdf(x, mu, sigma):

Example

Input

normal_pdf(0.0, mu=0.0, sigma=1.0)

Output

np.float64(0.3989422804014327)

Reasoning

At the mean, the exponent is 0, so the density is 1 / sqrt(2 * pi), about 0.3989.

Constraints and notes

• x may be a scalar or an array. Return the same shape.