Can you please help with udnerstanding the mathematical statistics question and solution attached? If possible, can you please provide an explanation and background on some of the ideas presented in the question and solution? Question:In Section 10.2.1, it was claimed that the random variables I(−∞,x](Xi) are independent. Why is this so?Solution:https://www.slader.com/textbook/9780534399429-mathematical-statistics-and-data-analysis-3rd-edition/408/exercises/4/ Please do not reject question if it is too complex for you and you do not know how to solve this problem.Allow another expert a chance to answer this question. In Section 10.2.1, it was claimed that the random variables 1-.) (X;) are inde-pendent. Why is this so?X1,..., Xn are independent 10.2 Methods Based on the CumulativeDistribution Function10.2.1 The Empirical Cumulative Distribution FunctionSuppose that x1, ..., x, is a batch of numbers. (The word sample is often used in thecase that the x; are independently and identically distributed with some distributionfunction; the word batch will imply no such commitment to a stochastic model.) Theempirical cumulative distribution function (ecdf) is defined as1F, (x)= - (#x; < x)(With this definition, F, is right-continuous; in the former Soviet Union and EasternEurope, the ecdf is usually defined to be left-continuous.)Denote the ordered batch of numbers by x(1) < x(2) <· · ·< x (n). Then if x < x(1),F,(x) = 0, if x(1) хF„(x)where(-00,x)(X;)ni=1The random variables I-o,x](X;) are independent Bernoulli random variables:I(-0.x) (X;) = {S 1,with probability F(x)0,with probability 1 – F(x)Thus, n F, (x) is a binomial random variable (n trials, probability F(x) of success)and soE[F,(x)] = F(x)%3DVar[F„(x)] =–F(x)[1 – F(x)]As an estimate of F(x), F„(x) is unbiased and has a maximum variance at that valueof x such that F (x)small, the variance tends to zero..5, that is, at the median. As x becomes very large or veryIn the preceding paragraph, we considered F„(x) for fixed x; the results can beapplied to form a confidence interval for F(x) for any given value of x. Much deeperanalysis focuses on the stochastic behavior of F, as a random function; that is, allvalues of x are considered simultaneously. It turns out, somewhat surprisingly, thatthe distribution ofmax |F„(x) – F (x)|-00

Name: Can you please help with udnerstanding the mathematical statistics que
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Description: Can you please help with udnerstanding the mathematical statistics question and solution attached? If possible, can you please provide an explanation and background on some of the ideas presented in the question and solution? Question:In Section 10.2