Probability Distributions and their Stories

• Story. We perform \(N\) Bernoulli trials, each with probability \(θ\) of success. The number of successes, \(n\), is Binomially distributed.
• Example. Distribution of plasmids between daughter cells in cell division. Each of the \(N\) plasmids as a chance \(θ\) of being in daughter cell 1 ("success"). The number of plasmids, \(n\), in daughter cell 1 is binomially distributed.
• Parameters. There are two parameters: the probability \(θ\) of success for each Bernoulli trial, and the number of trials, \(N\).
• Support. The Binomial distribution is supported on the set of nonnegative integers.
• Probability mass function.
  
  
  \(\begin{align}
  f(n;N,\theta) = \begin{pmatrix}
  N \\
  n
  \end{pmatrix}
  \theta^n (1-\theta)^{N-n}
  \end{align}\).
  
  
  
• Usage
  
  

  Package	Syntax
  NumPy	np.random.binomial(N, theta)
  SciPy	scipy.stats.binom(N, theta)
  Stan	binomial(N, theta)

  
  
• Related distributions.
  • The Bernoulli distribution is a special case of the Binomial distribution where \(N=1\).
• In the limit of \(N→∞\) and \(θ→0\) such that the quantity \(Nθ\) is fixed, the Binomial distribution becomes a Poisson distribution with parameter \(Nθ\).
• The Binomial distribution is a limit of the Hypergeometric distribution. Considering the Hypergeometric distribution and taking the limit of \(a+b→∞\) such that \(a/(a+b)\) is fixed, we get a Binomial distribution with parameters \(N=N\) and \(θ=a/(a+b)\).

params = [dict(name='N', start=1, end=20, value=5, step=1),
          dict(name='θ', start=0, end=1, value=0.5, step=0.01)]
app = distribution_plot_app(x_min=0,
                            x_max=20,
                            scipy_dist=st.binom,
                            params=params,
                            x_axis_label='n',
                            title='Binomial')
bokeh.io.show(app, notebook_url=notebook_url)