For this exercise, the reader must have access to a statistical package that
obtains the binomial distribution. Hints are given for \(\mathrm{R}\) code, but
other packages can be used too.
(a) Obtain the plot of the pmf for the \(b(15,0.2)\) distribution. Using
\(\mathrm{R}\), the following commands return the plot:
\(x<-0: 15 ;\) plot \(\left(\operatorname{dbinom}(x, 15, .2)^{-} x\right)\)
(b) Repeat part (a) for the binomial distributions with \(n=15\) and with \(p=\)
\(0.10,0.20, \ldots, 0.90 .\) Comment on the shapes of the pmf's as \(p\)
increases. Use the following \(\mathrm{R}\) segment:
\(\mathrm{x}<-0: 15 ; \quad\) par \((\mathrm{mfrow}=\mathrm{c}(3,3)) ;
\mathrm{p}<-1: 9 / 10\)
for \((j\) in \(p)\left\\{\right.\) plot \(\left(\right.\) dbinom \(\left.(x, 15,
j)^{\sim} x\right) ;\) title(paste \(\left.\left.(" p=", j)\right)\right\\}\)
(c) Let \(Y=\frac{X}{n}\), where \(X\) has a \(b(n, 0.05)\) distribution. Obtain the
plots of the pmfs of \(Y\) for \(n=10,20,50,200 .\) Comment on the plots (what do
the plots seem to be converging to as \(n\) gets large? ).