Question

Continuing with Exercise \(3.2 .8\), make a page of four overlay plots for the following 4 Poisson and binomial combinations: \(\lambda=2, p=0.02 ; \lambda=10, p=0.10\); \(\lambda=30, p=0.30 ; \lambda=50, p=0.50 .\) Use \(n=100\) in each situation. Plot the subset of the binomial range that is between \(n p \pm \sqrt{n p(1-p)} .\) For each situation, comment on the goodness of the Poisson approximation to the binomial.

Short answer

This exercise involves calculating binomial and Poisson probabilities for given parameters, creating overlay plots and evaluating the Poisson Approximation visually. The goodness of the Poisson approximation to the binomial can be subjectively determined by how closely the two plotted distributions match for each parameter set.

Step-by-step solution

Calculate Binomial Probabilities

First, we need to calculate the binomial probabilities for each tuple of \( \lambda, p \) combinations i.e., \( (\lambda=2, p=0.02), (\lambda=10, p=0.10), (\lambda=30, p=0.30), (\lambda=50, p=0.50) \) for binomial distribution B(n, p) where \( n=100 \).

Calculate Poisson Probabilities

Next, we need to calculate the Poisson probabilities over the subset of the binomial range that is between \( np \pm \sqrt{np(1-p)} \) for each \( \lambda \).

Create Overlay Plots

After calculating binomial and poisson probabilities, create an overlay plot for each set of binomial and Poisson probabilities using a suitable plotting tool.

Evaluate Poisson Approximation

Finally, assess the goodness of Poisson approximation by visually comparing plots of binomial and Poisson probabilities. If the Poisson plots follow closely with the Binomial plots, then the Poisson approximation can be considered 'good'.