Question

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? ).

Short answer

By generating multiple plots of the binomial distribution with R, one can see how the shape of the distribution changes with different parameter values. As \(p\) increases, the peak of the binomial distribution shifts right, implying more successes are more likely. When plotting the ratio of the number of successes to the number of trials \(Y\), one gets a distribution that centers around the probability of success, demonstrating the Law of Large Numbers.

Step-by-step solution

Understanding the Binomial Distribution

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.

Generating the Plot with R

To generate the plot with R, first, a sequence of x-values ranging from 0 to 15 is created using the command \(x<-0:15\). The pmf (probability mass function) of a binomial distribution can be obtained using the dbinom function in R. Thus, the plot can be created using the command \(\text{plot(dbinom(x, 15, 0.2)\(~\)x)}\).

Replicating with Different Parameters

To replicate the plot for binomial distributions with \(n=15\) and \(p=0.10, 0.20, ..., 0.90\), an R segment could be used. First, a sequence of probabilities is created. For each of these probabilities, plot the probability mass function of a binomial distribution with \(n=15\) and that probability. par(\(mfrow=c(3,3)\)) is used to arrange the multiple plots in a 3x3 grid.

Interpreting the Plots

As \(p\) increases, the most likely number of successes (the peak of the distribution) increases as well. This makes sense, as a higher probability of success means it is more likely to see more successes.

Generating Plots for the Distribution of \(Y\)

Similar steps would be followed to generate the plots of the probability mass functions of \(Y\) for different \(n\) values. Only this time \(Y=X/n\), where \(X\) follows a \(b(n, 0.05)\) distribution. This is the ratio of the number of successes to the total number of trials.

Interpreting the Plots of \(Y\)

As \(n\) gets large, the distribution of \(Y\) appears to approach a certain distribution, which is around \(p=0.05\), as this is the probability of success. This is due to the Law of Large Numbers, which states that as the number of trials gets larger, the sample mean (in this case \(Y\)) tends to get closer to the true mean (which is \(p\)).