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)