Simulation is an effective way for students to gain understanding of the Central Limit Theorem. MAPLE can do simulations. But MAPLE brings the added capability of being able to illustrate the Central Limit Theorem theoretically.

Let Yequal the sum of nrolls of an m-sided die. It is not difficult to find the p.d.f. of Ybut it can be tedious if nis very large. MAPLE can do this very easily. Graphically the probability histogram of Ycan be compared to a normal p.d.f. with mean and variance . Animation is an effective way to illustrate the convergence to normality as nincreases.

Let Yequal the sum of a random sample of size nfrom a continuous uniform distribution on the interval (0, 1). MAPLE can find the p.d.f. of Yand use animation to illustrate the convergence of the distribution of Yto a normal distribution.

Now let Yequal the sum of a random sample of size nfrom the distribution with p.d.f.

, < x< 1. Again MAPLE can find the p.d.f. of Y. It is extremely interesting to use animation to illustrate the convergence to normality because the p.d.f. of Yhas relative maxima. To help students understand why the p.d.f. of Yhas so many relative maxima, it is instructive to look at 3-D graphs. For example, let and let . Graph the joint p.d.f. of Uand V. And then graphically illustrate the integration needed to find the p.d.f. of , the sum of four U-shaped random variables.

To see some examples, click on one of the following two lines for choices.

Sums of Discrete Random Variables

Sums of Continuous Random Variables

To download this Maple worksheet, click here.

To download the extra Maple procedures, click here.