Let X1,., Xn be independent and identically distributed random variables with Var(X1) < o. Show that n 1 E;X; →p EX1. п(п + 1) j=1 Note. A simple way to solve a problem of showing Yn p a is to establish lim, EYn = a and lim, Var(Yn) = 0.

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The National Longitudinal Study of Adolescent Health interviewed several thousand teens (grades 7 to 12). One question asked was “What do you think are the chances you will be married in the next 10 years?” Here is a two-way table of the responses by gender:   Opinion Female Male Almost no chance 120 94 Some chance but probably not 156 178 A 50-50 chance 427 498 A good chance 750 700 Almost certain 1176 770   How many individuals are described in this table? How many females were among the respondents? The percent of females among the respondents was about %. Your percent from the previous exercise is part of the marginal distribution of opinion about marriage. the conditional distribution of sex among adolescents with a given opinion. the marginal distribution of sex. What percent of females thought that they were almost certain to be married in the next 10 years?% Your percent from the previous exercise is part of the conditional distribution of sex among…

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