4.1. "If two random variables are statistically independent, the coefficient of correlation between the two is zero. But the converse is not necessarily true; that is, zero correlation does not imply statistical independence. However, if two variables are normally distributed, zero correlation necessarily implies statistical independence." Verify this statement for the following joint probability density function of two normally distributed variables Y1 and Y2 (this joint probability density function is known as the bivariate normal probability density function):
4See App. A for a general discussion of the properties of the maximum likelihood estimators as well as for the distinction between asymptotic unbiasedness and consistency. Roughly speaking, in asymptotic unbiasedness we try to find out the lim E(an) as n tends to infinity, where n is the sample size on which the estimator is based, whereas in consistency we try to find out how an behaves as n increases indefinitely. Notice that the unbiasedness property is a repeated sampling property of an estimator based on a sample of given size, whereas in consistency we are concerned with the behavior of an estimator as the sample size increases indefinitely.
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