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Plotting Distributions with Python

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
In [ ]:
#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[ ]:
xs(x)sin(x)e^x
0-3.000000NaN-0.1411200.049787
1-2.993994NaN-0.1470630.050087
2-2.987988NaN-0.1530010.050389
3-2.981982NaN-0.1589340.050692
4-2.975976NaN-0.1648610.050998

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

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

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

In [ ]:
#import combination function
from scipy.special import comb
In [ ]:
#3 choose 2
comb(3, 2)
Out[ ]:
3.0
In [ ]:
#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]
In [ ]:
#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>