python - Why does scipy.norm.pdf sometimes give PDF > 1? How to correct it?

11
votes

Given mean and variance of a Gaussian (normal) random variable, I would like to compute its probability density function (PDF).

enter image description here

I referred this post: Calculate probability in normal distribution given mean, std in Python,

Also the scipy docs: scipy.stats.norm

But when I plot a PDF of a curve, the probability exceeds 1! Refer to this minimum working example:

import numpy as np
import scipy.stats as stats

x = np.linspace(0.3, 1.75, 1000)
plt.plot(x, stats.norm.pdf(x, 1.075, 0.2))
plt.show()

This is what I get:

Gaussian PDF Curve

How is it even possible to have 200% probability to get the mean, 1.075? Am I misinterpreting anything here? Is there any way to correct this?

2
pythonscipydistributionnormal-distribution
I actually did, @talonmies. The norm.pdf by itself is used for standardized random variables, hence it calculates exp(-x**2/2)/sqrt(2*pi). To bring mu and sigma into the relation, loc and and scale are introduced respectively. Specifying these would mean replacing x with (x-loc)/scale and dividing the final result by scale thus forming the Gaussian PDF as prescribed above. - Ébe Isaac

2 Answers

22
votes

It's not a bug. It's not an incorrect result either. Probability density function's value at some specific point does not give you probability; it is a measure of how dense the distribution is around that value. For continuous random variables, the probability at a given point is equal to zero. Instead of p(X = x), we calculate probabilities between 2 points p(x1 < X < x2) and it is equal to the area below that probability density function. Probability density function's value can very well be above 1. It can even approach to infinity.

0
votes

it's a density function, not a mass function

if variance is less than 1/(2*pi), the gaussian will exceed 1.0

exceeding 1 is only a limitation for mass functions, not density functions