The Binomial Distribution

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The Binomial Distribution#

Suppose that we are dealing with an experiment with two outcomes 0 (faillure) and 1 (success) and that the probability of success is \(\theta\). We are interested in the random variable \(X\) tha counts the number of successful experiments in \(n\) trials. This variable is called a Binomial random variable. We write:

\[ X\sim \text{Binomial}(n, \theta). \]

It can be shown (but beyond the scope of this class), that the probability of \(k\) successful experiments is given by the PMF:

\[ p(X = k) = {n\choose{k}}\theta^k(1-\theta)^{n-k}, \]

where \({n\choose{k}}\) is the number of \(k\) combinations out of \(n\) elements, i.e.:

\[ {n\choose{k}} = \frac{n!}{k!(n-k)!}. \]

Here is how to define the binomial in scipy.stats:

Hide code cell source 
import matplotlib.pyplot as plt
%matplotlib inline
import seaborn as sns
sns.set(rc={"figure.dpi":100, 'savefig.dpi':300})
sns.set_context('notebook')
sns.set_style("ticks")
from IPython.display import set_matplotlib_formats
set_matplotlib_formats('retina', 'svg')
import numpy as np
import scipy.stats as st

n = 5       # Performing the experiment n times
theta = 0.6 # Probability of sucess each time
X = st.binom(n, theta) # Number of successes

Here are some samples:

X.rvs(100)

array([4, 4, 3, 3, 3, 3, 2, 5, 4, 5, 4, 3, 3, 2, 4, 3, 5, 2, 1, 3, 4, 4,
       2, 4, 5, 3, 3, 5, 4, 5, 4, 4, 2, 4, 3, 3, 3, 4, 4, 2, 4, 3, 3, 3,
       1, 4, 3, 0, 3, 2, 1, 3, 2, 4, 3, 3, 2, 3, 4, 2, 5, 4, 2, 3, 1, 2,
       0, 4, 5, 3, 3, 4, 2, 3, 5, 1, 4, 2, 4, 2, 4, 4, 4, 2, 5, 4, 2, 4,
       2, 3, 1, 4, 2, 3, 2, 2, 1, 2, 1, 5])

Let’s draw the PMF:

fig, ax = plt.subplots(dpi=100)
xs = range(n + 1) # You can have anywhere between 0 and n successes
ax.bar(xs, X.pmf(xs))
ax.set_xlabel('$x$')
ax.set_ylabel('$p(x)$');
../_images/befbb2e36fc1f4121366e74b2332e9518fa25110755ed4a054043d598b842873.svg

Questions#

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