What is the Expectation of a Ratio of Independent Random Variables?

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The discussion centers on calculating the expectation of the ratio y = x_1/(x_1 + ... + x_n) for identically distributed independent random variables x_1, x_2, ..., x_n taking values in (1, 2). Participants suggest using the hint E[(x_1 + ... + x_n)/(x_1 + ... + x_n)] = 1 to derive the expectation of y. The conversation explores the implications of the independence and identical distribution of the variables on the expectation calculation. The focus remains on the mathematical approach and reasoning behind finding the expected value of the ratio. Understanding this expectation is crucial for applications in probability and statistics.
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Let x_1, x_2, ..., x_n be identically distributed independent random variables, taking values in (1, 2). If y = x_1/(x_1 + ... + x_n), then what is the expectation of y?
 
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Hint: use E[(x_1+...+x_n)/(x_1+...+x_n)]=1.
 

Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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