Scaling and Shifting Scipy Distributions in Python

I am working with scipy.stats.beta to generate a Beta distribution probability density function.

You can see the expected result for the standard interval of [0,1] here:

I would like to create the distribution on the interval [-1,1] instead. For a,b = 0.5, 0.5 one would expect the "u" shaped curve to distribute with the mean at 0 and ending at [-1,1]. However, with std = 1 and mean = 0 I get the following:

Now, if I set mean = -1 and std = 2 then I get close to what the desired result is:

Why is the PDF not scaling / centering as expected using the standard deviation and mean inputs? I must not be understanding the controls for the distribution function.

from scipy.stats import beta
import matplotlib.pyplot as plt

fig,ax = plt.subplots(nrows=1, ncols=1)

#x = np.linspace(beta.ppf(0.01, a, b),beta.ppf(0.99, a, b), 100)
x = np.arange(-1.0,1.0,0.01)

a, b = 0.5,0.5 #2.31, 0.627
mean = -1
std = 2
#mean, var, skew, kurt = beta.stats(a, b, moments='mvsk')
pdf = beta.pdf(x, a, b, loc=mean, scale=std)

ax.plot(x, pdf,'r-', lw=5, alpha=0.6, label='beta pdf')

mean = np.nanmean(pdf[np.isfinite(pdf)]) 
std = np.nanstd(pdf[np.isfinite(pdf)])

From the docs:

> The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, beta.pdf(x, a, b, loc, scale) is identically equivalent to beta.pdf(y, a, b) / scale with y = (x - loc) / scale.

So when you are using scale = 1 and loc = 0, you are simply getting the distribution beta.pdf(x, a, b), which you can really verify.

The reason why you get the first plot is that you are trying to plot a Beta distribution over an x domain that contains zero and Beta(0, 0.5, 0.5) tends to Infinity.

Note that you don't see the distribution exploding at the right side of your plot because your x.max() == 0.99

The parameters loc and scale are not the mean and standard deviation of the distribution. They are translation and scaling parameters; see, e.g. "Location-scale family". For the beta distribution, loc is the left end of the support interval, and scale is its length.