1.1 Basics of probability theory

    import pandas

from matplotlib import pyplot
import numpy as np
import math
from scipy.optimize import curve_fit
from scipy.integrate import quad
import plotly.offline as py
import plotly.graph_objs as go
from ipywidgets import interact, FloatSlider, Layout

from code.common import draw_classic_axes, configure_plotting

configure_plotting()
  

We consider a random variable \(x\) which has a set of possible outcomes \(S = \{x_1, x_2, ...\}\).

# Defining variables
mu = 0
sigma = 0.05
sigma_range = np.arange(0.04, 0.5, 0.01)
sigma_length = len(sigma_range)
active = 1
x_range = np.linspace(-2, 2, 1000)
length = len(x_range)

# Create figure
fig = go.Figure()

def gauss(sigma, mu, x):
   return (1/np.sqrt(2*math.pi*sigma**2)) * math.e**(-(x-mu)**2/(2*sigma)**2)

for current_sigma in sigma_range:
    fig.add_trace(
        go.Scatter(
            visible = False,
            x = x_range,
            y = [gauss(current_sigma, mu, x) for x in x_range] ,
            mode = 'lines',
            line_color = 'blue',
            line_dash = 'dot',
            name = 'Gauss distribution sigma = ' + str(sigma),
            fill = 'tonextx',
            fillcolor = 'lightblue'
        ))

fig.update_xaxes(range=[-1, 1])
fig.update_yaxes(range=[0, 10]) 
fig.data[active].visible = True


# Creation of the aditional images
steps = []
for i in range(sigma_length):
    step = dict(
        method = "update",
        args = [{"visible": [False] * length}],
        value = str(sigma_range[i])
    )
    step["args"][0]["visible"][i] = True
    steps.append(step)

# Creating the slider
sliders = [dict(
    tickcolor = 'White',
    font_color = 'White',
    currentvalue_font_color = 'Black',
    active = active,
    name = r'Standard deviation',
    font_size = 16,
    currentvalue = {"prefix": r"Standard deviation: "},
    pad = {"t": 50},
    steps = steps,
)]

# Updating the images for each step
fig.update_layout(
    sliders = sliders,
)

for i in range(sigma_length):
    fig['layout']['sliders'][0]['steps'][i]['label'] = ' %.2f ' % sigma_range[i]


fig

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1. Data source: Wikipedia, mainly the CRC Handbook of Chemistry and Physics. ↩

2. The data in this plot is the same as what Einstein used, but the curve in this plot is improved compared to what Einstein did, see this blog post for the backstory. ↩