Given $X_{1},\ldots,X_{n}$ an iid sample of a standard normal distribution, let $M=M(X_1,\ldots,X_n)$ be the median of this sample. What is the value of$$\mathbb E\left[M(X_1,\ldots,X_n)\times\sum_{i=1}^n{X_i}\right]$$
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2$\begingroup$ What have you tried If the $X_i$ are independently normally distributed each with mean $\mu$ and variance $\sigma^2$ then $\sum\limits_{i=1}^n X_i$ is normally distributed with mean $n\mu$ and variance $n\sigma^2$. $\endgroup$– HenryCommented Oct 15 at 9:53
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2$\begingroup$ Hint: the distribution of $M$ equals the distribution of $-M$ (provided you use standard definitions of median when $n$ is even where you split the difference between the two middle values). $\endgroup$– whuber ♦Commented Oct 15 at 13:31
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2$\begingroup$ I do not understand the meaning of $M(\sum_{i=1}^n X_i)$ since $\sum_{i=1}^n X_i$ is a single (Normal) random variable $\endgroup$– Xi'anCommented Oct 15 at 19:27
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2$\begingroup$ @Xi'an Fair question--I interpreted this to mean the median of the distribution of the sum; but the hint continues to apply when it's intended to be the median of the $(X_i),$ as suggested by my parenthetical remark. $\endgroup$– whuber ♦Commented Oct 15 at 19:51
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1$\begingroup$ Might be useful stats.stackexchange.com/questions/83840/… $\endgroup$– kjetil b halvorsen ♦Commented Oct 16 at 0:48
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1 Answer
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Assuming the question is about $X_1, X_2, \dotsc, X_n$ iid $\mathcal{N}(\mu, \sigma^2)$ with $\bar{X}$ its arithmetic mean and $M=\text{Median}(X_1, X_2, \dotsc, X_n)$. Then you ask for $\DeclareMathOperator{\E}{\mathbb{E}} \E \{ n M \bar{X} \}$
I assume a self-study question, so a hint. Use the result at https://stats.stackexchange.com/a/125430/11887 and the double expectation theorem, and the answer will follow fast.
answered Oct 16 at 1:12