Discrete distributions

Bernoulli distribution

  • Story. A Bernoulli trial is an experiment that has two outcomes that can be encoded as success (\(y=1\)) or failure (\(y=0\)). The result \(y\) of a Bernoulli trial is Bernoulli distributed.
  • Example. Check to see if a given bacterium is competent, given that it has probability \(θ\) of being competent.
  • Parameter. The Bernoulli distribution is parametrized by a single value, \(θ\), the probability that the trial is successful.
  • Support. The Bernoulli distribution may be nonzero only for zero and one.
  • Probability mass function.

    \(\begin{align}
    f(y;\theta) = \left\{ \begin{array}{ccc}
    1-\theta & & y = 0 \\(0.5em]
    \theta & & y = 1.
    \end{array}
    \right.
    \end{align}\)

  • Usage

    • Package Syntax
    NumPy np.random.choice([0, 1], p=[1-theta, theta])
    SciPy scipy.stats.bernoulli(theta)
    Stan bernoulli(theta)

  • Related distributions.
    • The Bernoulli distribution is a special case of the Binomial distribution with \(N=1\).
params = [dict(name='θ', start=0, end=1, value=0.5, step=0.01)]
app = distribution_plot_app(x_min=0,
                            x_max=1,
                            scipy_dist=st.bernoulli,
                            params=params,
                            x_axis_label='y',
                            title='Bernoulli')
bokeh.io.show(app, notebook_url=notebook_url)