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Bernoulli Variable

Explore how to simulate Bernoulli variables by flipping a coin multiple times and recording successes. Understand the binomial distribution formula, compute theoretical probabilities, and compare these to experimental results. Learn to visualize cumulative probability functions and deepen your grasp of random variables in scientific computing.

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In the example below, we will flip a coin five times in a row and record how many times we obtain tails (varying from 0-5). We will be performing the experiment 1000 times.

We will then compute the cumulative probability, print the values to the screen and make a plot of the cumulative probability function using a bar graph.

Python 3.5
import numpy as np
import matplotlib.pyplot as plt
import numpy.random as rnd
N = 1000
tails = np.sum(rnd.randint(0, 1+1, (5, 1000)), axis=0)
counttails = np.zeros(6, dtype='int')
for i in range(6):
    counttails[i] = np.count_nonzero(tails == i)
prob = counttails / N
cum_prob = np.cumsum(prob)
print('probababilties:', prob)
print('cumulative probabilities:', cum_prob)
plt.bar(range(0, 6), cum_prob)
plt.xticks(range(0, 6))
plt.xlabel('number of tails in two flips')
plt.ylabel('cumulative probability');

Execute the code several times and see that the graph changes a bit every time.

Probability of a Bernoulli variable

In the example above, we computed the probability of a certain number of heads in five flips experimentally. But we can, of course, compute the value exactly by using a few simple formulas.

Consider the random variable YY, which is the outcome of an experiment with two possible values 00 and 11. Let pp be the probability of success,

p=P( ...