Distribution Review¶

OBJECTIVES

Review Plotting Functions
Review Combination Numbers
Review Binomial Distribution
Review Normal Distribution
Review center and spread of distribution

Plotting Functions¶

In [ ]:

#import libraries
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd

In [ ]:

#define a function
def f(x): return x**2

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#plot the function
x = np.linspace(-3, 3, 1000)
plt.plot(x, f(x))

Out[ ]:

[<matplotlib.lines.Line2D at 0x7f098a7e2a58>]

In [ ]:

#square roots, trigonometric, exponential, and subplots
fig, ax = plt.subplots(1, 3, figsize = (16, 4))
def s(x): return np.sqrt(x)
def sin(x): return np.sin(x)
def e(x): return np.e**x
ax[0].plot(x, s(x))
ax[1].plot(x, sin(x))
ax[2].plot(x, e(x));

/usr/local/lib/python3.6/dist-packages/ipykernel_launcher.py:3: RuntimeWarning: invalid value encountered in sqrt
  This is separate from the ipykernel package so we can avoid doing imports until

In [ ]:

#tables of values
functions = pd.DataFrame({'x': x, 's(x)': s(x), 'sin(x)': sin(x), 'e^x': e(x)})
functions.head()

/usr/local/lib/python3.6/dist-packages/ipykernel_launcher.py:3: RuntimeWarning: invalid value encountered in sqrt
  This is separate from the ipykernel package so we can avoid doing imports until

Out[ ]:

	x	s(x)	sin(x)	e^x
0	-3.000000	NaN	-0.141120	0.049787
1	-2.993994	NaN	-0.147063	0.050087
2	-2.987988	NaN	-0.153001	0.050389
3	-2.981982	NaN	-0.158934	0.050692
4	-2.975976	NaN	-0.164861	0.050998

Combination Numbers¶

In mathematics, a combination is a selection of items from a collection, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange. More formally, a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements, the number of k-combinations is equal to the binomial coefficient

\displaystyle {\binom {n}{k}}={\frac {n(n-1)\dotsb (n-k+1)}{k(k-1)\dotsb 1}},

which can be written using factorials as $\frac {n!}{k!(n-k)!}$ whenever $\displaystyle k\leq n$ , and which is zero when $k>n$ . The set of all k-combinations of a set S is often denoted by $\binom {S}{k}$ .

In [ ]:

#import combination function
from scipy.special import comb

In [ ]:

#3 choose 2
comb(3, 2)

Out[ ]:

3.0

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#3 choose 0, 1, 2, 3
[comb(3, i) for i in [0, 1, 2, 3]]

Out[ ]:

[1.0, 3.0, 3.0, 1.0]

In [ ]:

#build the triangle
for i in range(10):
  print([comb(i, j) for j in range(i + 1)])

[1.0]
[1.0, 1.0]
[1.0, 2.0, 1.0]
[1.0, 3.0, 3.0, 1.0]
[1.0, 4.0, 6.0, 4.0, 1.0]
[1.0, 5.0, 10.0, 10.0, 5.0, 1.0]
[1.0, 6.0, 15.0, 20.0, 15.0, 6.0, 1.0]
[1.0, 7.0, 21.0, 35.0, 35.0, 21.0, 7.0, 1.0]
[1.0, 8.0, 28.0, 56.0, 70.0, 56.0, 28.0, 8.0, 1.0]
[1.0, 9.0, 36.0, 84.0, 126.0, 126.0, 84.0, 36.0, 9.0, 1.0]

In [ ]:

#row 7
row_seven = [1.0, 7.0, 21.0, 35.0, 35.0, 21.0, 7.0, 1.0]

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#plot the row
plt.bar(range(8), row_seven)

Out[ ]:

<BarContainer object of 8 artists>

In [ ]:

#total
sum(row_seven)

Out[ ]:

128.0

In [ ]:

#probability of 3 successes
35/128

Out[ ]:

0.2734375

In [ ]:

#binomial distribution
#import 
import scipy.stats as stats
b = stats.binom(7, .5)
b.pmf(3)

Out[ ]:

0.27343750000000017

In [ ]:

#example
plt.bar(range(8), b.pmf(range(8)))

Out[ ]:

<BarContainer object of 8 artists>

Plotting Distributions with Python

Distribution Review¶

Plotting Functions¶

Combination Numbers¶