We begin by considering a random variable $X_k$ with Binomial distribution $B(k;p)$. This means that $P(X_k=r)=\left(\genfrac{}{}{0ex}{}{k}{r}\right)p^r(1-p{)}^{k-r}$.

sage: def binom(k,p,r):
...       return N(binomial(k,r)*p^r*(1-p)^(k-r))

We now plot the frequency plot of $X_k$ for a fixed $p$ and $k$ varying over some range.

sage: p=1/2
sage: binoms=[bar_chart(map(lambda r:binom(k,p,r),range(k+1)),color='gray',width=0.2)
sage: for k in range(4,31)]

We animate this sequence of plots.

sage: binanim=animate(binoms,xmin=0,xmax=21,ymin=0,ymax=0.5,figsize=[6,4])

Looking at all the charts together, we see that:

• Since the expectation $E(X_k)=kp$ keeps increasing with $k$, the centre of the plot moves to the right.
• The height of the peak decreases and the distribution "flattens out".

sage: binanim.show(delay=100)

Binomial flattens as k increases

We now take the random variable $V_k$ whose distribution is given by $B(k;c/k)$ for some fixed value of $c$. The mean of this sequence of distributions remains fixed and $V_k$ converge to a random variable $V$ which has the Poisson Distribution

$$P(V=r)=\frac{c^r{e}^{-c}}{r!}$$

sage: c=0.9
sage: pois=bar_chart(map(lambda r:c^r*exp(-c)/factorial(r),range(6)), width=0.3,color="orange") # This is the Poisson
sage: plseq=[pois+bar_chart(map(lambda r:binom(2*k,c/(2*k),r),range(6)),color='gray',width=0.15)
sage: for k in range(1,30)] # These are all the Binomials
sage: poisanim=animate(plseq,xmin=0,xmax=6,ymin=0,ymax=c/2,figsize=[6,4])

sage: poisanim.show(delay=70)

Binomial tends to Poisson