The second one is the Gaussian Distribution \begin{equation*} p(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp{(-\frac{(x-\mu)^2}{2\sigma^2})}, \end{equation*} with mean value \mu and standard deviation \sigma . If \mu=0 and \sigma=1 , it is normally called the standard normal distribution \begin{equation*} p(x) = \frac{1}{\sqrt{2\pi}} \exp{(-\frac{x^2}{2})}, \end{equation*}
The following simple Python code plots the above distribution for different values of \mu and \sigma .
xxxxxxxxxximport numpy as npfrom math import acos, exp, sqrtfrom matplotlib import pyplot as pltfrom matplotlib import rc, rcParamsimport matplotlib.units as unitsimport matplotlib.ticker as tickerrc('text',usetex=True)rc('font',**{'family':'serif','serif':['Gaussian distribution']})font = {'family' : 'serif', 'color' : 'darkred', 'weight' : 'normal', 'size' : 16, }pi = acos(-1.0)mu0 = 0.0sigma0 = 1.0mu1= 1.0sigma1 = 2.0mu2 = 2.0sigma2 = 4.0x = np.linspace(-20.0, 20.0)v0 = np.exp(-(x*x-2*x*mu0+mu0*mu0)/(2*sigma0*sigma0))/sqrt(2*pi*sigma0*sigma0)v1 = np.exp(-(x*x-2*x*mu1+mu1*mu1)/(2*sigma1*sigma1))/sqrt(2*pi*sigma1*sigma1)v2 = np.exp(-(x*x-2*x*mu2+mu2*mu2)/(2*sigma2*sigma2))/sqrt(2*pi*sigma2*sigma2)plt.plot(x, v0, 'b-', x, v1, 'r-', x, v2, 'g-')plt.title(r'{\bf Gaussian distributions}', fontsize=20)plt.text(-19, 0.3, r'Parameters: $\mu = 0$, $\sigma = 1$', fontdict=font)plt.text(-19, 0.18, r'Parameters: $\mu = 1$, $\sigma = 2$', fontdict=font)plt.text(-19, 0.08, r'Parameters: $\mu = 2$, $\sigma = 4$', fontdict=font)plt.xlabel(r'$x$',fontsize=20)plt.ylabel(r'$p(x)$ [MeV]',fontsize=20)# Tweak spacing to prevent clipping of ylabel plt.subplots_adjust(left=0.15)plt.savefig('gaussian.pdf', format='pdf')plt.show()