_05e_pascal.Pascal Class Reference

Class for generating and applying pascal triangles on some applications, such as probability and statistics. More...

Inheritance diagram for _05e_pascal.Pascal:

Collaboration diagram for _05e_pascal.Pascal:

Public Member Functions
def	__init__ (self, first_level, last_level, newline='\n', center=True)
	Create a binomial array as a list of lines. More...
def	binomialArray (self, n)
	Compute n lines of the binomial array. More...
def	binomial_coefficient (self, n, m, method=0)
	Compute \(n\) choose \(m\) or Combination \((n,m)\). More...
def	binomial_cumulative_distribution (self, p, n, x=None)
	Compute the probability of getting at most \(x\) successes in \(n\) trials, which is the sum of the first \(x\) terms of the binomial expansion \((p+q)^n\). More...
def	exponent (self, val, type=False)
	Return a superscript string for the given value. More...
def	binomial_expansion (self, e, html=False)
	Compute the binomial expansion of \((x+y)^n\). More...
def	getNextRow (self, curr_row)
	Computes the next row of a Pascal triangle, given the current row. More...
def	fmtRow (self, r, pad=True)
	Formats a row of the Pascal triangle. More...
def	__repr__ (self)
	Creates the pascal triangle between two levels. More...
def	__str__ (self)
	Creates the pascal triangle between two levels, using the binomial array. More...

Public Attributes
	first_level
	first level of the triangle to be shown. More...
	last_level
	last level of the triangle to be shown. More...
	center
	whether to center the lines of the triangle. More...
	lineFeed
	string used to signify the end of a line of text and the start of a new one. More...
	superscript
	translation table for creating superscript. More...
	subscript
	translation table for creating subscript. More...
	binomial_array
	Binomial array. More...

Detailed Description

Class for generating and applying pascal triangles on some applications, such as probability and statistics.

Constructor & Destructor Documentation

__init__()

def _05e_pascal.Pascal.__init__	(		self,
			first_level,
			last_level,
			newline = '\n',
			center = True
	)

Create a binomial array as a list of lines.

Member Function Documentation

__repr__()

def _05e_pascal.Pascal.__repr__

(

self

)

Creates the pascal triangle between two levels.

Returns

a string with the lines of this pascal triangle.

See also

https://www.cut-the-knot.org/arithmetic/combinatorics/PascalTriangleProperties.shtml

https://www.johndcook.com/blog/2016/07/05/distribution-of-numbers-in-pascals-triangle/

References _05e_pascal.Pascal.center, _05e_pascal.Pascal.first_level, _05e_pascal.Pascal.fmtRow(), _05e_pascal.Pascal.getNextRow(), _05e_pascal.Pascal.last_level, and _05e_pascal.Pascal.lineFeed.

__str__()

def _05e_pascal.Pascal.__str__

(

self

)

Creates the pascal triangle between two levels, using the binomial array.

The lines are centered in respect to the last line of the triangle.

Returns

a string with the lines of this pascal triangle.

References _05e_pascal.Pascal.binomial_array, _05e_pascal.Pascal.center, _05e_pascal.Pascal.first_level, _05e_pascal.Pascal.fmtRow(), _05e_pascal.Pascal.last_level, and _05e_pascal.Pascal.lineFeed.

binomial_coefficient()

def _05e_pascal.Pascal.binomial_coefficient	(		self,
			n,
			m,
			method = 0
	)

Compute \(n\) choose \(m\) or Combination \((n,m)\).

Parameters

n	number of objects to choose from.
m	number of places.
method	how to get the result.

Returns

\(C{{n}\choose{m}} = {{n!} \over {(n-m)!\ m!}} = {{n (n-1) ... (n-m+1)} \over {m!}}\), number of ways of choosing m objects from n.

See also

https://en.wikipedia.org/wiki/Combination

https://docs.scipy.org/doc/scipy/reference/tutorial/special.html

https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.comb.html#scipy.special.comb

References _05e_pascal.Pascal.binomial_array.

Referenced by _05e_pascal.Pascal.binomial_cumulative_distribution(), and _05e_pascal.Pascal.binomial_expansion().

binomial_cumulative_distribution()

def _05e_pascal.Pascal.binomial_cumulative_distribution	(		self,
			p,
			n,
			x = None
	)

Compute the probability of getting at most \(x\) successes in \(n\) trials, which is the sum of the first \(x\) terms of the binomial expansion \((p+q)^n\).

In other words, the probability of obtaining at most \(x\) successes in \(n\) independent trials, each of which has a probability \(p\) of success, and \(q = (1-p)\) of failure.

That is, if \(X\) denotes the number of successes, it returns:

• \(P(X \le x) = \sum_{i=0}^{x} {C {{n}\choose{i}}\ p^i\ (1-p)^{n-i}}\).

Parameters

p	probability of success.
n	number of trials.
x	number of successes.

Returns

the probability of getting at most x successes in n independent trials.

See also

https://www.khanacademy.org/math/ap-statistics/probability-ap/probability-multiplication-rule/v/getting-at-least-one-heads

http://www.wolframalpha.com/widgets/view.jsp?id=d821210668c6cc5a02db1069cc52464f

https://mat.iitm.ac.in/home/vetri/public_html/statistics/binomial.pdf

https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-random-variables/v/visualizing-a-binomial-distribution

References _05e_pascal.Pascal.binomial_coefficient().

binomial_expansion()

def _05e_pascal.Pascal.binomial_expansion	(		self,
			e,
			html = False
	)

Compute the binomial expansion of \((x+y)^n\).

For e = 5, return:

• \(x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5\) (html)
• x⁵ + 5x⁴y + 10x³y² + 10x²y³ + 5xy⁴ + y⁵ (unicode translation table)
• x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5 (None)

Parameters

e	exponent n.
html	exponent creation method. True for html.

Returns

a string representing the expansion.

See also

https://www.khanacademy.org/math/precalculus/prob-comb/prob-combinatorics-precalc/v/exactly-three-heads-in-five-flips

https://www.khanacademy.org/math/precalculus/prob-comb/combinations/v/combination-formula

https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/binomial-random-variables/v/binomial-distribution

References _05e_pascal.Pascal.binomial_coefficient(), and _05e_pascal.Pascal.exponent().

binomialArray()

def _05e_pascal.Pascal.binomialArray	(		self,
			n
	)

Compute n lines of the binomial array.

Complexity is \(1 + 2\ +\ ...\ +\ n\) elements. This is arithmetic progression that sums to \({n*(n+1)}\over{2}\), which is in \(O(n^2)\).

   0 - 1
   1 - 1 1
   2 - 1 2 1
   3 - 1 3 3 1
   4 - 1 4 6 4 1
   5 - 1 5 10 10 5 1
   

Parameters

n	number of lines.

exponent()

def _05e_pascal.Pascal.exponent	(		self,
			val,
			type = False
	)

Return a superscript string for the given value.

Parameters

val	a numeric string.
type	select the superscript method.

Returns

a new superscript string.

References _05e_pascal.Pascal.superscript.

Referenced by _05e_pascal.Pascal.binomial_expansion().

fmtRow()

def _05e_pascal.Pascal.fmtRow	(		self,
			r,
			pad = True
	)

Formats a row of the Pascal triangle.

Parameters

r	row to be printed.
pad	whether to pad the row with blanks to the left.

Returns

string representing the row.

References _05e_pascal.Pascal.last_level.

Referenced by _05e_pascal.Pascal.__repr__(), and _05e_pascal.Pascal.__str__().

getNextRow()

def _05e_pascal.Pascal.getNextRow	(		self,
			curr_row
	)

Computes the next row of a Pascal triangle, given the current row.

Parameters

curr_row

current row.

Returns

next_row: list(map(sum, zip([0] + curr_row, curr_row + [0])))

Referenced by _05e_pascal.Pascal.__repr__().

Member Data Documentation

binomial_array

_05e_pascal.Pascal.binomial_array

Binomial array.

Referenced by _05e_pascal.Pascal.__str__(), and _05e_pascal.Pascal.binomial_coefficient().

center

_05e_pascal.Pascal.center

whether to center the lines of the triangle.

Referenced by _05e_pascal.Pascal.__repr__(), and _05e_pascal.Pascal.__str__().

first_level

_05e_pascal.Pascal.first_level

first level of the triangle to be shown.

Referenced by _05e_pascal.Pascal.__repr__(), and _05e_pascal.Pascal.__str__().

last_level

_05e_pascal.Pascal.last_level

last level of the triangle to be shown.

Referenced by _05e_pascal.Pascal.__repr__(), _05e_pascal.Pascal.__str__(), and _05e_pascal.Pascal.fmtRow().

lineFeed

_05e_pascal.Pascal.lineFeed

string used to signify the end of a line of text and the start of a new one.

Referenced by _05e_pascal.Pascal.__repr__(), and _05e_pascal.Pascal.__str__().

subscript

_05e_pascal.Pascal.subscript

translation table for creating subscript.

superscript

_05e_pascal.Pascal.superscript

translation table for creating superscript.

Referenced by _05e_pascal.Pascal.exponent().

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The documentation for this class was generated from the following file:

• _05e_pascal.py