probably-probability

A python project for various probability-related explorations. Work with, sample, visualize, combine, compare all your favorite random variables, play poker, or take a random walk on the numberline.

Supported Distributions

Discrete

• Bernoulli
• Binomial
• Geometric
• Hyper geometric
• Poisson

Continuous

• Uniform
• Exponential
• Normal
• Erlang

Supported Functionalities

• Basic Functions
  • Expected value
  • PDF
  • CDF
  • Generic and recursive
  • Tail
  • Variance
  • Higher Moments [work in progress]
    • Via MGF, Laplace/z-Transform, or directly, depending on the distribution
• Simulation
  • Generate a random instance of a given distribution
  • Generic and recursive inverse transform method
  • Continuous accept/reject method
  • k rounds of generating independent instances of a given distribution
  • k independent rounds, j times
  • Compare sums of one distribution to other distributions
    • e.g: Erlang distribution as sum of exponentials
    • Binomials as sums of Bernoullis, etc.
• Visualization
  • Plotting
    • plot and compare any numerical function of the RandomVariable class with other distributions and/or other conditions of the same distribution
    • compare sampled data to the true distribution graphically
    • plot the pdf,cdf,tail etc.
    • visualize the max of binomials across k trials
• Random Walks [work in progress]
  • Visualize 1D random walks, and some properties of them [not completely committed yet (plot.py)]
• Poker [work in progress]
  • Play a random poker hand
  • Play k random poker hands, and obtain a bar chart of the probabilities of obtaining each outcome
  • See the poker directory for another README

Examples

Basics of Working with Distributions

• Print the PDF of a Geometric Random Variable

  X = Geometric(0.3)
  print(X.pdf(5))
  > 0.072029

• Compute the fourth moment of the Exponential distribution

  • setting verbose to true will print the intermediate derivatives, using the Laplace transform in the case of the Exponential distribution

    X = Exponential(.1)
    print(X.moment(4,verbose=True)) # should be 4!/(.1^4) = 240000
    > 0.1/(x + 0.1)
    > -0.1/(x + 0.1)**2
    > 0.2/(x + 0.1)**3
    > -0.6/(x + 0.1)**4
    > 240000.000000000

• Bound the probability that a Binomial distribution deviates from its mean by at least a certain value using Chebyshev's inequality

  X = Binomial(100,.5)
  print(u.chebyshevs(X,25)) # P[|X - E[X]| >= 25] 
  > 0.04

• Compare the PDFs of different binomial distributions

P = Plot() P.plot({'binomial':([(20,.3),(20,.5),(20,.7)],20,1)},'pdf')

<img width="813" alt="Screenshot 2023-04-17 at 11 43 37 PM" src="https://user-images.githubusercontent.com/85849407/232665569-ff264196-53b9-43a5-ad0f-434ee1518e45.png">

#### Sampling

* ##### Sample a random instance of the Normal distribution
```python
X = Normal(-10,10)
print(X.genVar())
> -12.483660083014579

• Sample some distributions for 10000 iterations

util = Util() util.simAll(k=10000)

<img width="813" alt="Screenshot 2023-04-17 at 10 35 53 PM" src="https://user-images.githubusercontent.com/85849407/232655813-d59862ef-ec0c-4e70-a595-a00b7af2459e.png">

* ##### Sample Poisson(10) 10000 times and compare the percentages of outcomes to the pmf graphically
```python

P = Plot()
P.plotSamples(Poisson(10),10000)

Random Walks

• Visualize simple bounded 1D random walks

rw = NumberLineRW() rw.graphWalks(100,50)

<img width="611" alt="Screenshot 2023-04-30 at 9 03 46 PM" src="https://user-images.githubusercontent.com/85849407/235386663-228a76f9-7b83-4f28-b1e2-5f0ca91532a1.png">
* Visualize simple bounded 2D random walks

```python

rw = IntegerLatticeRW()
rw.graphWalks2D(100,50)

• You can vary p as well:

rw = IntegerLatticeRW(.7)
rw.graphWalks2D(100,50)