The following problems are taken from Montgomery & Runger, Applied Statistics and Probability for Engineers, 2nd edition.
4-57, 4-59, 4-103, 4-107, 4-112, 4-115
5-120, 5-122, 5-128
6-110
(1) Using MINITAB, Excel, R, or some other software, plot a graph to show the Bin(72, 1/36) distribution and its approximating Poisson distribution. Plot another graph to show the Bin(72, 1/36) distribution and its approximating Normal distribution.
[Hint: See last year's A02 solutions for some suggestions of how to do this. You can, however, do better than that. Last year I said I didn't know how to plot the Binomial with bars and the Normal with a continuous line on the same graph in MINITAB. Now I do. Compute the Binomial pmf over a grid of x-values; 0 to 10 is OK for this problem. Compute the approximating Normal pdf over a finer grid. In Plot, give the Binomial columns as Y-X values, with "Data Display" set to "Project"; then, in "Annotation", give the Normal X and Y values as a "Line Annotation". I did the Poisson pmf similarly, using the same x-grid as the Binomial, but as a "Marker Annotation" with open circles. (When specifying the "Points" in "Annotation", I found it only worked if I gave the X and Y values as column numbers separated by a space, e.g. "C3 C4".)]
(2) Let X be the sum of 5 fair 6-sided dice. Find the theoretical mean and variance of X. Will the distribution be approximately Normal? Verify empirically by rolling 5 dice 20 times, or by simulating 1000 rolls of 5 dice using random discrete uniform data in MINITAB. Compute the sample mean and variance and plot the data on a histogram with a normal curve superimposed.
[Hint: In MINITAB, the simplest way to draw a histogram with a normal curve superimposed is to use "Stat > Basic Statistics > Display Descriptive Statistics > Graphs > Histogram of data, with normal curve". To simulate a discrete uniform distribution for the roll of one die, you need to specify the distribution by setting up a column with integers 1 to 6 and a column with the corresponding probabilities, i.e. 1/6 at each point. To simulate 1000 rolls of 5 dice, put the random single-die scores into 5 columns, each with 1000 rows, just like we did with the normal and exponential data in Exercise 2.]