2 , ..., 12. 3.45 Simulate a Prussian soldier's death by horse kick as in Example 3.8. Create a histogram based on 10,000 repetitions. Compare to a histogram of the Poisson distribution using the command rpois (10000 , 0 . 61).A collection of random variables, such as an infinite sequence, is independent if for all finite subgroups X1, ..., Xx of the collection, P(X1 = 11, . . ., Xk = [k) = P(X1 = 11) . . . P(Xk = Ik), for all I1, . . ., Ik. Example 3.8 The Current Population Survey of 2010 provides data on the propor- tion of family households in the United States by number of children under 18 years old. (See Table 3.2). In a sample of four households, what is the probability that no household has children? Although the actual sampling was done without replacement, since the popula- tion is so large, we assume sampling with replacement and independence, as the difference is negligible. For k = 1, 2, 3, 4, let Xx be the number of children for the kth family household in the sample. Then the desired probability is P(X1 = 0, X2 = 0, X3 = 0, X4 = 0) = P(X1 = 0) P(X2 = 0) P(X3 = 0)P(X4 = 0) = (0.5533)* = 0.0938. TABLE 3.2: Distribution of number of children in U.S. households. 0 2 3 or more 0.5533 0.1922 0.1642 0.0903