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Chapter 2 2.1 These outcomes are random because they are not known with certainty until they actually occur. You do not know with certainty the gender of the next person you will meet, the time that it will take to commute to school, and so forth. 2.2 If X and Y are independent, then Pr(Y ≤ y | X = x) = Pr(Y ≤ y) for all values of y and x. That is, independence means that the conditional and marginal distributions of Y are identical so that learning the value of X does not change the probability distribution of Y: Knowing the value of X says nothing about the probability that Y will take on different values. 2.3 Although there is no apparent causal link between rainfall and the number of children born, rainfall could tell you something about the number of children born. Knowing the amount of rainfall tells you something about the season, and births are seasonal. Thus, knowing rainfall tells you something about the month, which tells you something about the number of children born. Thus, rainfall and the number of children born are not independently distributed. 2.4 The average weight of four randomly selected students is unlikely to be exactly 145 lbs. Different groups of four students will have different sample average weights, sometimes greater than 145 lbs. and sometimes less. Because the four students were selected at random, their sample average weight is also random. 2.5 All of the distributions will have a normal shape and will be centered at 1, the mean of Y. However they will have different spreads because they have different variances. The variance of Y is 4/n, so the variance shrinks as n gets larger. In your plots, the spread of the normal density when n = 2 should be wider than when n = 10, which should be wider than when n = 100. As n gets very large, the variance approaches zero, and the normal density collapses around the mean of Y. That is, the distribution of Y becomes highly concentrated around Y as n grows large (the probability that Y is close to Y tends to 1), which is just what the law of large numbers says. 2.6 The normal approximation does not look good when n = 5, but looks good for n = 25 and n =100. Thus Pr( Y ≤ 0.1) is approximately equal to the value computed by the normal approximation when n is 25 or 100, but is not well approximated by the normal distribution when n = 5. 2.7 The probability distribution looks liked Figure 2.3b, but with more mass concentrated in the tails. Because the distribution is symmetric around Y = 0, Pr(Y > c) = Pr(Y < c) and, because this is substantial mass in the tails of the distribution, Pr(Y > c) remains significantly greater than zero even for large values of c. 1