Question at position 11 What is the expected value of the sum of 20 independent and identically distributed squared standard normal random variables?AnswerWhat is the expected value of the sum of 20 independent and identically distributed squared standard normal random variables?[input]BlackTom题目解析简答题
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Question at position 11 What is the expected value of the sum of 20 independent and identically distributed squared standard normal random variables?AnswerWhat is the expected value of the sum of 20 independent and identically distributed squared standard normal random variables?[input]BlackTom题目解析简答题

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BlackTom AI · Accepted Answer

The question asks for the expected value of the sum of 20 independent and identically distributed squared standard normal random variables.
If Z1,...,Z20 are iid N(0,1), then Zi^2 are chi-square(1) variables. The sum S = sum_{i=1}^{20} Zi^2 follows a chi-square distribution with 20 degrees of freedom (Chi-square(20)).
A key property of the chi-square distribution is that its expected value equals its degrees of freedom, i.e., E[S] = 20.
Equivalently, since each Zi^2 has expectation 1 (because Zi has mean 0 and variance 1, E[Zi^2] = Var(Zi) = 1), the linearity of expectation gives E[S] = sum E[Zi^2] = 20 * 1 = 20.
Note on the provided options: there are no answer choices listed in the prompt (the answer_options field is empty), so there is nothing to compare against. The calculation above directly yields the expected value regardless of listed options.

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Question at position 8 Let X be a chi-squared random variable with degrees of freedom 14. What is P(X=14)?AnswerLet X be a chi-squared random variable with degrees of freedom 14. What is P(X=14)?[input]BlackTom题目解析

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