What does this mean exactly?

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I'm a bit confused about the part with z-score. It seems to be important when testing hypothesis.

  • $\begingroup$ What is the original source of this? $\endgroup$ – QuantIbex Aug 9 '13 at 9:20

The notation '$X_k \in N(\mu,\sigma)$' in your source means that the random variable $X_k$ has a normal distribution with mean $\mu$ and standard deviation $\sigma$. The second symbol in $N(\cdot,\cdot)$ denotes the standard deviation, NOT the variance. This was brought to my attention by @Dilip Sarwate (see comment below).

It is more common to see the notation '$Y \sim N(u,\sigma^2)$' where the symbol '$\sim$' is prefered to '$\in$', which is more relevant to indicate membership of an element in a set, and where the second symbol in $N(\cdot,\cdot)$ denotes the variance. However, some softwares/programming languages also use standard deviation instead of variance as the second parameter (e.g. functions [dpqr]norm in R).

  • 1
    $\begingroup$ No!! In the notation of the source, $N(\mu,\sigma)$ means a normal distribution with mean $\mu$ and standard deviation $\sigma$, not variance $\sigma$ as you state. Note that the variance of the sum of $n$ independent identically distributed random variables with common variance $A$ (and stddev $\sqrt{A}$) is $nA$ while the standard deviation of the sum is $\sqrt{nA} = \sqrt{A}\cdot\sqrt{n} = \operatorname{stddev}\cdot\sqrt{n}$. $\endgroup$ – Dilip Sarwate Aug 8 '13 at 21:02
  • $\begingroup$ @Dilip Sarwate, thanks for spoting this! I've amended the answer accordingly. $\endgroup$ – QuantIbex Aug 8 '13 at 22:22
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    $\begingroup$ I cannot see how $\sim$ could be equated with $\in$: the former names a random variable whose distribution is determined by a particular tuple of parameters $(u,\sigma^2)$ whereas the latter is a generic element of a set of distributions. I suppose you could (rigorously) write "$Y\in\{X|X\sim \mathcal{N}(u,\sigma^2)\}$"--but that even more clearly shows the distinction between $\sim$ ("has a distribution given by") and $\in$ ("is an element of"). $\endgroup$ – whuber Aug 9 '13 at 2:24

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