Section 2.2 — Multiple random variables¶

This notebook contains all the code examples from Section 2.2 Multiple random variables in the No Bullshit Guide to Statistics.

Notebook setup¶

In [1]:

# Ensure required Python modules are installed
%pip install --quiet numpy scipy seaborn

# Ensure required Python modules are installed %pip install --quiet numpy scipy seaborn

[notice] A new release of pip is available: 26.0.1 -> 26.1.1
[notice] To update, run: pip install --upgrade pip
Note: you may need to restart the kernel to use updated packages.

In [2]:

# load Python modules
import matplotlib.pyplot as plt  # generic plotting functions
import numpy as np               # numerical calculations
import pandas as pd              # data frames to store joint PMFs
import seaborn as sns            # plotting distributions

# load Python modules import matplotlib.pyplot as plt # generic plotting functions import numpy as np # numerical calculations import pandas as pd # data frames to store joint PMFs import seaborn as sns # plotting distributions

In [3]:

# Figures setup
plt.clf()  # needed otherwise `sns.set_theme` doesn't work
sns.set_theme(
    context="paper",
    style="whitegrid",
    palette="colorblind",
    rc={"font.family": "serif",
        "font.serif": ["Palatino", "DejaVu Serif", "serif"],
        "figure.figsize": (5,3)},
)
%config InlineBackend.figure_format = 'retina'

# Figures setup plt.clf() # needed otherwise `sns.set_theme` doesn't work sns.set_theme( context="paper", style="whitegrid", palette="colorblind", rc={"font.family": "serif", "font.serif": ["Palatino", "DejaVu Serif", "serif"], "figure.figsize": (5,3)}, ) %config InlineBackend.figure_format = 'retina'

<Figure size 640x480 with 0 Axes>

In [4]:

# Simple float __repr__  
import numpy as np
if int(np.__version__.split(".")[0]) >= 2:
    np.set_printoptions(legacy='1.25')

# set random seed for repeatability
np.random.seed(42)

# Simple float __repr__ import numpy as np if int(np.__version__.split(".")[0]) >= 2: np.set_printoptions(legacy='1.25') # set random seed for repeatability np.random.seed(42)

Definitions¶

In [5]:

pXY = [[ 0.1, 0.05, 0.02, 0.01, 0.01],
       [0.06, 0.12, 0.06, 0.03, 0.03],
       [0.02, 0.06, 0.12, 0.07, 0.02],
       [0.01, 0.01, 0.05,  0.1, 0.05]]
jpmfXY = pd.DataFrame(data=pXY,
                      index=["a", "b", "c", "d"],
                      columns=[1,2,3,4,5])
jpmfXY.columns.name = "X"
jpmfXY.index.name = "Y"
# print(jpmfXY.sum().sum())
jpmfXY

pXY = [[ 0.1, 0.05, 0.02, 0.01, 0.01], [0.06, 0.12, 0.06, 0.03, 0.03], [0.02, 0.06, 0.12, 0.07, 0.02], [0.01, 0.01, 0.05, 0.1, 0.05]] jpmfXY = pd.DataFrame(data=pXY, index=["a", "b", "c", "d"], columns=[1,2,3,4,5]) jpmfXY.columns.name = "X" jpmfXY.index.name = "Y" # print(jpmfXY.sum().sum()) jpmfXY

Out[5]:

X	1	2	3	4	5
Y
a	0.10	0.05	0.02	0.01	0.01
b	0.06	0.12	0.06	0.03	0.03
c	0.02	0.06	0.12	0.07	0.02
d	0.01	0.01	0.05	0.10	0.05

In [6]:

from ministats.plots.probability import plot_joint_pmf_stems
ax = plot_joint_pmf_stems(jpmfXY.T, zmax=0.2)
# ax.view_init(elev=30, azim=-30)
# ax.set_box_aspect((4, 5, 3))

from ministats.plots.probability import plot_joint_pmf_stems ax = plot_joint_pmf_stems(jpmfXY.T, zmax=0.2) # ax.view_init(elev=30, azim=-30) # ax.set_box_aspect((4, 5, 3))

In [7]:

from ministats.plots.probability import plot_joint_pmf_balloons
plot_joint_pmf_balloons(jpmfXY);

from ministats.plots.probability import plot_joint_pmf_balloons plot_joint_pmf_balloons(jpmfXY);

In [8]:

print(sum(jpmfXY.loc["b"]))
print(jpmfXY.loc["b"].values)
print(jpmfXY.loc["b"].values / sum(jpmfXY.loc["b"]))

print(sum(jpmfXY.loc["b"])) print(jpmfXY.loc["b"].values) print(jpmfXY.loc["b"].values / sum(jpmfXY.loc["b"]))

0.3
[0.06 0.12 0.06 0.03 0.03]
[0.2 0.4 0.2 0.1 0.1]

In [9]:

from ministats.book.tables import joint_pmf_to_array
joint_pmf_to_array(jpmfXY, sigfigs=3)

from ministats.book.tables import joint_pmf_to_array joint_pmf_to_array(jpmfXY, sigfigs=3)

Out[9]:

$$ \begin{array}{c|ccccc} f_{XY} & X=1 & X=2 & X=3 & X=4 & X=5 \\ \hline Y=a & 0.1 & 0.05 & 0.02 & 0.01 & 0.01 \\ Y=b & 0.06 & 0.12 & 0.06 & 0.03 & 0.03 \\ Y=c & 0.02 & 0.06 & 0.12 & 0.07 & 0.02 \\ Y=d & 0.01 & 0.01 & 0.05 & 0.1 & 0.05 \\ \end{array} $$

Marginal distributions¶

Conditional distributions¶

In [ ]:

Joint probability distributions¶

Example 1: two coin tosses¶

In [10]:

def fC(c):
    if c in ["heads", "tails"]:
        return 1/2
    else:
        return 0

def fC1C2(c1,c2):
    return fC(c1) * fC(c2)


for c1 in ["heads", "tails"]:
    for c2 in ["heads", "tails"]:
        print("Pr(" + c1 + "," + c2 + ") =", fC1C2(c1,c2))

def fC(c): if c in ["heads", "tails"]: return 1/2 else: return 0 def fC1C2(c1,c2): return fC(c1) * fC(c2) for c1 in ["heads", "tails"]: for c2 in ["heads", "tails"]: print("Pr(" + c1 + "," + c2 + ") =", fC1C2(c1,c2))

Pr(heads,heads) = 0.25
Pr(heads,tails) = 0.25
Pr(tails,heads) = 0.25
Pr(tails,tails) = 0.25

In [11]:

import pandas as pd

#        C1=heads    C1=tails
pC1C2 = [[0.25,      0.25   ],   # C2=heads
         [0.25,      0.25   ]]   # C2=tails

jpmfC1C2 = pd.DataFrame(data=pC1C2,
                        columns=["heads", "tails"],
                        index=["heads", "tails"])
jpmfC1C2.columns.name = "C_1"
jpmfC1C2.index.name = "C_2"

import pandas as pd # C1=heads C1=tails pC1C2 = [[0.25, 0.25 ], # C2=heads [0.25, 0.25 ]] # C2=tails jpmfC1C2 = pd.DataFrame(data=pC1C2, columns=["heads", "tails"], index=["heads", "tails"]) jpmfC1C2.columns.name = "C_1" jpmfC1C2.index.name = "C_2"

In [12]:

from ministats.book.tables import joint_pmf_to_array
joint_pmf_to_array(jpmfC1C2, sigfigs=3, margins=True)

from ministats.book.tables import joint_pmf_to_array joint_pmf_to_array(jpmfC1C2, sigfigs=3, margins=True)

Out[12]:

$$ \begin{array}{c|cc|c} f_{C_1C_2} & C_1=\texttt{heads} & C_1=\texttt{tails} & f_{C_2} \\ \hline C_2=\texttt{heads} & 0.25 & 0.25 & 0.5 \\ C_2=\texttt{tails} & 0.25 & 0.25 & 0.5 \\ \hline f_{C_1} & 0.5 & 0.5 & \\ \end{array} $$

In [13]:

from ministats.plots.probability import plot_joint_pmf_stems

plot_joint_pmf_stems(jpmfC1C2, zmax=0.3);
ax.text2D(1.15, 0.56, "$f_{C_1C_2}$", transform=ax.transAxes,
          fontsize=plt.rcParams["axes.labelsize"],
          rotation=90, va="center", ha="center");

from ministats.plots.probability import plot_joint_pmf_stems plot_joint_pmf_stems(jpmfC1C2, zmax=0.3); ax.text2D(1.15, 0.56, "$f_{C_1C_2}$", transform=ax.transAxes, fontsize=plt.rcParams["axes.labelsize"], rotation=90, va="center", ha="center");

In [14]:

from ministats.plots.probability import plot_joint_pmf_balloons
plot_joint_pmf_balloons(jpmfC1C2);

from ministats.plots.probability import plot_joint_pmf_balloons plot_joint_pmf_balloons(jpmfC1C2);

Example 2: rolling two dice¶

In [15]:

def fD(d):
    if d in [1,2,3,4,5,6]:
        return 1/6
    else:
        return 0

def fD1D2(d1,d2):
    return fD(d1) * fD(d2)

def fD(d): if d in [1,2,3,4,5,6]: return 1/6 else: return 0 def fD1D2(d1,d2): return fD(d1) * fD(d2)

In [16]:

#         D1=1  D1=2  D1=3  D1=4  D1=5  D1=6
pD1D2 = [[1/36, 1/36, 1/36, 1/36, 1/36, 1/36],  # D2=1
         [1/36, 1/36, 1/36, 1/36, 1/36, 1/36],  # D2=2
         [1/36, 1/36, 1/36, 1/36, 1/36, 1/36],  # D2=3
         [1/36, 1/36, 1/36, 1/36, 1/36, 1/36],  # D2=4
         [1/36, 1/36, 1/36, 1/36, 1/36, 1/36],  # D2=5
         [1/36, 1/36, 1/36, 1/36, 1/36, 1/36]]  # D2=6
jpmfD1D2 = pd.DataFrame(data=pD1D2,
                        columns=[1,2,3,4,5,6],
                        index=[1,2,3,4,5,6])
jpmfD1D2.columns.name = "D_1"
jpmfD1D2.index.name = "D_2"

# D1=1 D1=2 D1=3 D1=4 D1=5 D1=6 pD1D2 = [[1/36, 1/36, 1/36, 1/36, 1/36, 1/36], # D2=1 [1/36, 1/36, 1/36, 1/36, 1/36, 1/36], # D2=2 [1/36, 1/36, 1/36, 1/36, 1/36, 1/36], # D2=3 [1/36, 1/36, 1/36, 1/36, 1/36, 1/36], # D2=4 [1/36, 1/36, 1/36, 1/36, 1/36, 1/36], # D2=5 [1/36, 1/36, 1/36, 1/36, 1/36, 1/36]] # D2=6 jpmfD1D2 = pd.DataFrame(data=pD1D2, columns=[1,2,3,4,5,6], index=[1,2,3,4,5,6]) jpmfD1D2.columns.name = "D_1" jpmfD1D2.index.name = "D_2"

In [17]:

from ministats.plots.probability import plot_joint_pmf_stems
plot_joint_pmf_stems(jpmfD1D2.T, zmax=0.045);

from ministats.plots.probability import plot_joint_pmf_stems plot_joint_pmf_stems(jpmfD1D2.T, zmax=0.045);

In [18]:

from ministats.plots.probability import plot_joint_pmf_balloons
plot_joint_pmf_balloons(jpmfD1D2);

from ministats.plots.probability import plot_joint_pmf_balloons plot_joint_pmf_balloons(jpmfD1D2);

In [19]:

sum_to_seven = [(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)]
plot_joint_pmf_balloons(jpmfD1D2, highlight=sum_to_seven);

sum_to_seven = [(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)] plot_joint_pmf_balloons(jpmfD1D2, highlight=sum_to_seven);

Marginal distribution functions¶

In [20]:

from ministats.plots.probability import plot_joint_pmf_and_marginals
plot_joint_pmf_and_marginals(jpmfXY);

from ministats.plots.probability import plot_joint_pmf_and_marginals plot_joint_pmf_and_marginals(jpmfXY);

Conditional distributions¶

In [21]:

from ministats.plots.probability import plot_joint_pmf_and_conditional
plot_joint_pmf_and_conditional(jpmfXY, given="y");

from ministats.plots.probability import plot_joint_pmf_and_conditional plot_joint_pmf_and_conditional(jpmfXY, given="y");

In [22]:

plot_joint_pmf_and_conditional(jpmfXY, given="x");

plot_joint_pmf_and_conditional(jpmfXY, given="x");

Example 3: coin-dependent dice roll¶

In [23]:

def fC(c):
    if c in ["heads", "tails"]:
        return 1/2
    else:
        return 0

def fD6(d):
    if d in [1,2,3,4,5,6]:
        return 1/6
    else:
        return 0

def fD4(d):
    if d in [1,2,3,4]:
        return 1/4
    else:
        return 0

def fDC(d,c):
    if c == "heads":
        return fC(c) * fD6(d)
    elif c == "tails":
        return fC(c) * fD4(d)

for c in ["heads", "tails"]:
    for d in [1,2,3,4,5,6]:
        print("Pr(" + str(d) + "," + c + ") =", round(fDC(d,c),4))

def fC(c): if c in ["heads", "tails"]: return 1/2 else: return 0 def fD6(d): if d in [1,2,3,4,5,6]: return 1/6 else: return 0 def fD4(d): if d in [1,2,3,4]: return 1/4 else: return 0 def fDC(d,c): if c == "heads": return fC(c) * fD6(d) elif c == "tails": return fC(c) * fD4(d) for c in ["heads", "tails"]: for d in [1,2,3,4,5,6]: print("Pr(" + str(d) + "," + c + ") =", round(fDC(d,c),4))

Pr(1,heads) = 0.0833
Pr(2,heads) = 0.0833
Pr(3,heads) = 0.0833
Pr(4,heads) = 0.0833
Pr(5,heads) = 0.0833
Pr(6,heads) = 0.0833
Pr(1,tails) = 0.125
Pr(2,tails) = 0.125
Pr(3,tails) = 0.125
Pr(4,tails) = 0.125
Pr(5,tails) = 0.0
Pr(6,tails) = 0.0

In [24]:

sum(fDC(d,c) for d in [1,2,3,4,5,6] for c in ["heads", "tails"])

sum(fDC(d,c) for d in [1,2,3,4,5,6] for c in ["heads", "tails"])

Out[24]:

1.0

In [25]:

# f_{Y|X}(y|heads)  [as a list]
[fDC(d,"heads") for d in range(1,6+1)]

# f_{Y|X}(y|heads) [as a list] [fDC(d,"heads") for d in range(1,6+1)]

Out[25]:

[0.08333333333333333,
 0.08333333333333333,
 0.08333333333333333,
 0.08333333333333333,
 0.08333333333333333,
 0.08333333333333333]

In [26]:

# f_{Y|X}(y|tails)  [as a list]
[fDC(d,"tails") for d in range(1,6+1)]

# f_{Y|X}(y|tails) [as a list] [fDC(d,"tails") for d in range(1,6+1)]

Out[26]:

[0.125, 0.125, 0.125, 0.125, 0.0, 0.0]

In [27]:

# f_Y(y)    [as a list]
fDs = [fDC(d,"heads")+fDC(d,"tails") for d in range(1,6+1)]
fDs

# f_Y(y) [as a list] fDs = [fDC(d,"heads")+fDC(d,"tails") for d in range(1,6+1)] fDs

Out[27]:

[0.20833333333333331,
 0.20833333333333331,
 0.20833333333333331,
 0.20833333333333331,
 0.08333333333333333,
 0.08333333333333333]

In [28]:

# ALT. compute f_{D|C}(d|heads)*0.5 + f_{D|C}(d|tails)*0.5
fDs = [fD6(d)*0.5+fD4(d)*0.5 for d in range(1,6+1)]
fDs

# ALT. compute f_{D|C}(d|heads)*0.5 + f_{D|C}(d|tails)*0.5 fDs = [fD6(d)*0.5+fD4(d)*0.5 for d in range(1,6+1)] fDs

Out[28]:

[0.20833333333333331,
 0.20833333333333331,
 0.20833333333333331,
 0.20833333333333331,
 0.08333333333333333,
 0.08333333333333333]

In [29]:

sum(fDs)

sum(fDs)

Out[29]:

0.9999999999999999

In [30]:

import pandas as pd
#       D=1  D=3  D=3  D=4  D=5  D=6
pDC = [[1/6, 1/6, 1/6, 1/6, 1/6, 1/6],  # C=heads
       [1/4, 1/4, 1/4, 1/4,   0,   0]]  # C=tails
jpmfDC = pd.DataFrame(data=pDC,
                      index=["heads","tails"],
                      columns=[1,2,3,4,5,6])
jpmfDC.columns.name = "D"
jpmfDC.index.name = "C"

import pandas as pd # D=1 D=3 D=3 D=4 D=5 D=6 pDC = [[1/6, 1/6, 1/6, 1/6, 1/6, 1/6], # C=heads [1/4, 1/4, 1/4, 1/4, 0, 0]] # C=tails jpmfDC = pd.DataFrame(data=pDC, index=["heads","tails"], columns=[1,2,3,4,5,6]) jpmfDC.columns.name = "D" jpmfDC.index.name = "C"

In [31]:

from ministats.book.tables import joint_pmf_to_array
joint_pmf_to_array(jpmfDC, sigfigs=3)

from ministats.book.tables import joint_pmf_to_array joint_pmf_to_array(jpmfDC, sigfigs=3)

Out[31]:

$$ \begin{array}{c|cccccc} f_{DC} & D=1 & D=2 & D=3 & D=4 & D=5 & D=6 \\ \hline C=\texttt{heads} & 0.167 & 0.167 & 0.167 & 0.167 & 0.167 & 0.167 \\ C=\texttt{tails} & 0.25 & 0.25 & 0.25 & 0.25 & 0 & 0 \\ \end{array} $$

In [32]:

from ministats.plots.probability import plot_joint_pmf_stems
ax = plot_joint_pmf_stems(jpmfDC.T)

from ministats.plots.probability import plot_joint_pmf_stems ax = plot_joint_pmf_stems(jpmfDC.T)

In [33]:

from ministats.plots.probability import plot_joint_pmf_balloons
plot_joint_pmf_balloons(jpmfDC);

from ministats.plots.probability import plot_joint_pmf_balloons plot_joint_pmf_balloons(jpmfDC);

Example 4: diagnostic test¶

In [34]:

def fV(v):
    if v == 1:
        return 0.03
    elif v == 0:
        return 0.97

def fTgivenV1(t):
    if t == 1:
        return 0.90
    elif t == 0:
        return 0.10

def fTgivenV0(t):
    if t == 1:
        return 0.20
    elif t == 0:
        return 0.80

def fTgivenV(t,v):
    if v == 1:
        return fTgivenV1(t)
    elif v == 0:
        return fTgivenV0(t)

def fTV(t,v):
    return fTgivenV(t,v) * fV(v)

for v in range(0,1+1):
    for t in range(0,1+1):
        print("Pr({T=" + str(t) + ",V=" + str(v)+"})",
              "=", fTgivenV(t,v), "*", fV(v),
              "=", round(fTV(t,v),4))

def fV(v): if v == 1: return 0.03 elif v == 0: return 0.97 def fTgivenV1(t): if t == 1: return 0.90 elif t == 0: return 0.10 def fTgivenV0(t): if t == 1: return 0.20 elif t == 0: return 0.80 def fTgivenV(t,v): if v == 1: return fTgivenV1(t) elif v == 0: return fTgivenV0(t) def fTV(t,v): return fTgivenV(t,v) * fV(v) for v in range(0,1+1): for t in range(0,1+1): print("Pr({T=" + str(t) + ",V=" + str(v)+"})", "=", fTgivenV(t,v), "*", fV(v), "=", round(fTV(t,v),4))

Pr({T=0,V=0}) = 0.8 * 0.97 = 0.776
Pr({T=1,V=0}) = 0.2 * 0.97 = 0.194
Pr({T=0,V=1}) = 0.1 * 0.03 = 0.003
Pr({T=1,V=1}) = 0.9 * 0.03 = 0.027

In [35]:

import pandas as pd

#              T = 0           T = 1 
pTV = [[       0.8*0.97,  (1-0.8)*0.97],    #   V = 0
        [  (1-0.9)*0.03,      0.9*0.03]]    #   V = 1

jpmfTV = pd.DataFrame(data=pTV, columns=[0,1], index=[0,1])
jpmfTV.columns.name = "T"
jpmfTV.index.name = "V"

import pandas as pd # T = 0 T = 1 pTV = [[ 0.8*0.97, (1-0.8)*0.97], # V = 0 [ (1-0.9)*0.03, 0.9*0.03]] # V = 1 jpmfTV = pd.DataFrame(data=pTV, columns=[0,1], index=[0,1]) jpmfTV.columns.name = "T" jpmfTV.index.name = "V"

In [36]:

plot_joint_pmf_stems(jpmfTV.T);

plot_joint_pmf_stems(jpmfTV.T);

In [37]:

from ministats.plots.probability import plot_joint_pmf_balloons
plot_joint_pmf_balloons(jpmfTV, size_exponent=1);

from ministats.plots.probability import plot_joint_pmf_balloons plot_joint_pmf_balloons(jpmfTV, size_exponent=1);

In [38]:

from ministats.book.tables import joint_pmf_to_array
joint_pmf_to_array(jpmfTV, sigfigs=3)

from ministats.book.tables import joint_pmf_to_array joint_pmf_to_array(jpmfTV, sigfigs=3)

Out[38]:

$$ \begin{array}{c|cc} f_{TV} & T=0 & T=1 \\ \hline V=0 & 0.776 & 0.194 \\ V=1 & 0.003 & 0.027 \\ \end{array} $$

In [39]:

joint_pmf_to_array(jpmfTV, sigfigs=3, margins=True)

joint_pmf_to_array(jpmfTV, sigfigs=3, margins=True)

Out[39]:

$$ \begin{array}{c|cc|c} f_{TV} & T=0 & T=1 & f_{V} \\ \hline V=0 & 0.776 & 0.194 & 0.97 \\ V=1 & 0.003 & 0.027 & 0.03 \\ \hline f_{T} & 0.779 & 0.221 & \\ \end{array} $$

In [42]:

# f_T(0) = 
0.8 * 0.97 + 0.10 * 0.03

# f_T(0) = 0.8 * 0.97 + 0.10 * 0.03

Out[42]:

0.779

In [43]:

# f_T(1) = 
0.2 * 0.97 + 0.90 * 0.03

# f_T(1) = 0.2 * 0.97 + 0.90 * 0.03

Out[43]:

0.221

In [44]:

for t in range(0,1+1):
    fTt = fTV(t,0) + fTV(t,1)
    print("Pr({T="+str(t)+"}) =", fTt)

for t in range(0,1+1): fTt = fTV(t,0) + fTV(t,1) print("Pr({T="+str(t)+"}) =", fTt)

Pr({T=0}) = 0.779
Pr({T=1}) = 0.221

Bayes' rule¶

Example 4: diagnostic test (continued)¶

We're now interested in $f_{V|T}(v|t)$, which we can obtain using Bayes' theorem.

$$ f_{V|T}(v|t) = \frac{ f_{T|V}(t|v) f_V(v) }{ f_{T|V}(t|0) f_V(0) + f_{T|V}(t|1) f_V(1)}. $$

We'll now define the function fVgivenT that computes the numerator and denominator in this fraction, and returns the ratio.

In [45]:

def fVgivenT(v,t):
    num = fTgivenV(t,v) * fV(v)
    denom = fTgivenV(t,0) * fV(0) + fTgivenV(t,1) * fV(1)
    return num / denom

def fVgivenT(v,t): num = fTgivenV(t,v) * fV(v) denom = fTgivenV(t,0) * fV(0) + fTgivenV(t,1) * fV(1) return num / denom

The probability of the patient having the virus $f_{V|T}(1|t=1)$, given they tested positive is:

In [46]:

fVgivenT(1,1)

fVgivenT(1,1)

Out[46]:

0.12217194570135746

Intuitively, this percentage is measuring the ratio of the probability of the outcome $f_{VT}(1,1)$ divided by sum of the probabilities of all outcomes where positive test can occur: $f_{VT}(1,1)+f_{VT}(0,1)$.

In [47]:

fTV(1,1) / (fTV(1,0) + fTV(1,1))

fTV(1,1) / (fTV(1,0) + fTV(1,1))

Out[47]:

0.12217194570135746

Multivariable expectations¶

Covariance¶

Correlation¶

Independent random variables¶

Formulas for independent random variables¶

Independent, identically distributed variables¶

Example 5: tossing a coin $n$ times¶

Example 6: rolling a die $n$ times¶

Discussion¶

Dependence and independence¶

Exercises¶

See the notebook sec22_multiple_discrete_RVs.ipynb.

Links¶

In [48]:

fD(4) * fC("heads") / fDs[3]

fD(4) * fC("heads") / fDs[3]

Out[48]:

0.4

In [ ]: