4.1.1 Suppose that X1, X2, X3 are i.i.d. from px in Example 4.1.1. Determine the exact distribution of Y 3= (X1X2X3)1/3.
Below is example 4.1.1
previous table. Directl tly computing Py, as we have done for py2, would be onerous even for a computer! So what can we do here? One possibility is to look at the distribution of Y (Xi .- Xn)1/n when n is large and see if we can approximate this in some fashion. The results of Section 4.4.1 show that 71 has an approximate normal distribution when n is large. In fact, the approximating nor mal distribution when n = 20 turns out to be an N (0.447940, 0.105167) distribution. We have plotted this density in Figure 4.1.1 Another approach is to use the methods of Section 2.10 to generate N samples of size 20 from px, calculate In Y2o for each (ln is a 1-1 transformation, and we transform to avoid the potentially large values assumed by Y20), and then use these N values to approximate the distribution of In Y20- For example, in Figure 4.1.2 we have provided a plot of a density histogram (see Section 5.4.3 for more discussion of histograms) of N = 104 values of In Y20 calculated from N = lợ samples of size n = 20 generated (using the computer) from px- The area of each rectangle corresponds to the proportion of values of In Y2o that were in the interval given by the base of the rectangle. As we will see in Sections 4.2, 4.3, and 4.4, these areas approximate the actual probabilities that In Y20 falls in these intervals. These approximations improve as we increase N Notice the similarity in the shapes of Figures 4.1.1 and 4.1.2. Figure 4.1.2 is not symmetrical about its center, however, as it is somewhat skewed. This is an indication that the normal approximation is not entirely adequate when n 20.I