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The differences between the two-sided geometric distribution and the "standard" geometric distribution are:

1. The samples/results can include 0 and negative numbers, and
2. the "probability of staying at 0" is calculated differently than the other integers.

The Python code (below) implements Example 2.1 from the OP's link. Specifically, given the probability mass function for a two-sided geometric distribution: $$Pr[ Z = z] = \frac{1-\alpha}{1+\alpha}\alpha^{\left | z \right |}$$

• zeroProb() implements $\frac{1-\alpha}{1+\alpha}$
• np.random.geometric(1-p) implements $\alpha^{\left | z \right |}$
• signProb() accounts for the absolute value in $\alpha^{\left | z \right |}$

    #!/usr/bin/env python
    
    import random
    import numpy as np
    
    # The "chance of staying put at 0" is the main difference from the
    # standard implementation of a geometric distribution.
    
    # zeroProb() will return 1 if you "leave 0," which effectively
    # "turns ON" the standard implementation in twoSidedGeoDist().
    def zeroProb(p):
      #                    this is the chance of "staying put at 0"
      if random.random() < (1.0 - p)/(1.0 + p):
        return 0
      else:
        return 1
    
    # Coin flip to determine if the result is negative or positive.
    # This only applies when we "leave 0."
    def signProb():
      if random.random() < 0.5:
        return -1
      else:
        return 1
    
    def twoSidedGeoDist(p):
      #      (1) Did we "leave 0"? [Y=1|N=0]          (3) +/-
      return zeroProb(p) * np.random.geometric(1-p) * signProb()
    #                      (2) "Leave 0" i.e. Standard implementation
    
    alpha = 1.0/3.0
    counts = {}
    trials = 1000000
    
    for i in range(trials):
      result = twoSidedGeoDist(alpha)
      if result in counts:
        counts[result] = counts[result] + 1
      else:
        counts[result] = 1
    
    for x in sorted(counts.keys()):
      print(str(x) + "\t" + str(counts[x]) + "\t" + str(counts[x]*1.0 / trials))