This code is taken from https://matplotlib.org/examples/animation/index.html and extended for the normally distributed case.

%matplotlib inline

from scipy.stats import beta
import time
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation

class UpdateDistbeta(object):
    def __init__(self, ax, prob=0.5):
        self.success = 0
        self.prob = prob
        self.line, = ax.plot([], [], 'k-',color='blue')
        self.data_point = ax.text(0.02, 0.95, '', transform=ax.transAxes)
        self.success_rate = ax.text(0.02, 0.90, '', transform=ax.transAxes)
        self.bayes_estimator = ax.text(0.02, 0.85, '', transform=ax.transAxes)
        self.x = np.linspace(0, 1, 200)
        self.ax = ax

        # Set up plot parameters
        self.ax.set_xlim(0, 1)
        self.ax.set_ylim(0, 15)
        self.ax.grid(True)

        # This vertical line represents the theoretical value, to
        # which the plotted distribution should converge.
        self.ax.axvline(prob, linestyle='--', color='red')

    def init(self):
        self.success = 0
        self.line.set_data([], [])
        return self.line,

    def __call__(self, i):
        # This way the plot can continuously run and we just keep
        # watching new realizations of the process
        if i == 0:
            return self.init()

        # Choose success based on exceed a threshold with a uniform pick
        data = 0
        if np.random.rand(1,) < self.prob:
            self.success += 1
            data = 1
        y = beta.pdf(self.x, self.success + 1, (i - self.success) + 1)
        self.line.set_data(self.x, y)
        rate = self.success/float(i)
        bayes_estimator = (self.success+1)/(i+2)
        self.data_point.set_text('Data point = %i' % data)
        self.success_rate.set_text('MLE for success rate = %.5f' % rate)
        self.bayes_estimator.set_text('Bayes estimator = %.5f' % bayes_estimator)
        return self.line,

# Fixing random state for reproducibility
np.random.seed(19680801)


fig, ax = plt.subplots()
ud = UpdateDistbeta(ax, prob=0.3)
animbeta = FuncAnimation(fig, ud, frames=np.arange(100), init_func=ud.init,
                     interval=100, blit=True)
animbeta.save('moviebeta.mp4')

from scipy.stats import norm
import time
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation

class UpdateDistnormal(object):
    def __init__(self, ax, mu0, sigma0, sigma):
        self.mu0 = mu0
        self.sigma0 = sigma0
        self.sigma = sigma 
        self.data = np.random.normal(self.mu0,self.sigma0,1)
        self.cumdata = 0
        self.line, = ax.plot([], [], 'k-',color='blue')
        self.data_point = ax.text(0.02, 0.95, '', transform=ax.transAxes)
        self.cumdata_point = ax.text(0.02, 0.90, '', transform=ax.transAxes)
        self.true_value = ax.text(0.02, 0.85, '', transform=ax.transAxes)
        self.bayes_estimator = ax.text(0.02,0.80, '',transform=ax.transAxes)
        self.x = np.linspace(-10, 10, 1000)
        self.ax = ax

        # Set up plot parameters
        self.ax.set_xlim(-10, 10)
        self.ax.set_ylim(0, 5)
        self.ax.grid(True)
        self.ax.axvline(self.data, linestyle='--', color='red')

        # This vertical line represents the theoretical value, to
        # which the plotted distribution should converge.
        

    def init(self):
        self.line.set_data([], [])
        return self.line,

    def __call__(self, i):
        # This way the plot can continuously run and we just keep
        # watching new realizations of the process
        if i == 0:
            return self.init()

        # Choose success based on exceed a threshold with a uniform pick
        data = np.random.normal(self.data,self.sigma,1)
        self.cumdata = self.cumdata + data
        sigmaupdate = (self.sigma0**(-2)+self.sigma**(-2))**(-0.5)
        self.mu0 = (data/self.sigma**2+self.mu0/self.sigma0**2)*sigmaupdate**2
        self.sigma0 = sigmaupdate
        y = norm.pdf(self.x, self.mu0, self.sigma0)
        self.line.set_data(self.x, y)
        average = self.cumdata/i
        self.data_point.set_text('Data point = %.5f' % data)
        self.cumdata_point.set_text('MLE  = %.5f' % average)
        self.true_value.set_text('True value = %.5f' % self.data)
        self.bayes_estimator.set_text('Bayes estimator = %.5f' % self.mu0)
        return self.line,

# Fixing random state for reproducibility
np.random.seed(19680802)


fig, ax = plt.subplots()
ud = UpdateDistnormal(ax, mu0 = 5.0, sigma0 = 1.0, sigma = 5.0)
animnormal= FuncAnimation(fig, ud, frames=np.arange(100), init_func=ud.init,
                     interval=100, blit=True)
animnormal.save('movienormal.mp4')