from __future__ import division
%matplotlib inline
import pylab as pl
import numpy as np

xs = np.random.normal(size=1000000)
sigmas = np.linspace(.01, 10)
xs_sigs = xs * sigmas[:, np.newaxis]

pl.figure(figsize=(8, 3))

pl.subplot(121)
pl.plot(sigmas, np.cos(xs_sigs).mean(axis=1))
pl.plot(sigmas, np.exp(- sigmas**2 / 2), color='red', linestyle='--')

pl.subplot(122)
pl.plot(sigmas, np.sin(xs_sigs).mean(axis=1))
pl.tight_layout()

pl.figure(figsize=(6, 2))

pl.subplot(121)
pl.plot(sigmas, np.cos(xs_sigs).std(axis=1))
pl.plot(sigmas, (1 - np.exp(- sigmas**2)) / np.sqrt(2), color='red', linestyle='--')

pl.subplot(122)
pl.plot(sigmas, np.sin(xs_sigs).std(axis=1))
pl.plot(sigmas, np.sqrt((1 - np.exp(-2 * sigmas**2)) / 2), color='red', linestyle='--')
pl.tight_layout()

xs = np.random.normal(size=1000000)
mus = np.linspace(-5, 5)
sigmas = 1
xs_mus = xs + mus[:, np.newaxis]

pl.figure(figsize=(8, 3))

pl.subplot(121)
pl.plot(mus, np.cos(xs_mus).mean(axis=1))
pl.plot(mus, np.exp(- sigmas**2 / 2) * np.cos(mus), color='red', linestyle='--')

pl.subplot(122)
pl.plot(mus, np.sin(xs_mus).mean(axis=1))
pl.plot(mus, np.exp(- sigmas**2 / 2) * np.sin(mus), color='red', linestyle='--')
pl.tight_layout()

pl.figure(figsize=(8, 3))

pl.subplot(121)
pl.plot(mus, np.cos(xs_mus).std(axis=1))
pl.plot(mus, np.sqrt(.5 + np.exp(-2*sigmas**2) / 2 * np.cos(2 * mus) - np.exp(-sigmas**2) * np.cos(mus)**2),
        color='red', linestyle='--')

pl.subplot(122)
pl.plot(mus, np.sin(xs_mus).std(axis=1))
pl.plot(mus, np.sqrt(.5 - np.exp(-2*sigmas**2) / 2 * np.cos(2 * mus) - np.exp(-sigmas**2) * np.sin(mus)**2),
        color='red', linestyle='--')
pl.tight_layout()