Probability Distributions and their Stories

• 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)