with respect to $\sigma$, where $\sigma$ is the standard deviation of the standardarized random variable $x$ and $\mu$ its mean. I guess that it is


Correct me if I am wrong.

  • 1
    $\begingroup$ Didn't you forget to differentiate $(m-d-\mu)/\sigma$ with respect to $\sigma$? $\endgroup$ – yohBS Sep 27 '12 at 7:38
  • $\begingroup$ @yohBS Sorry, I did that. There is $(1/σ^2)$ multiplying my answer. Apart from that, do you think it is correct? $\endgroup$ – Daniel Lårs Sep 27 '12 at 7:46
  • $\begingroup$ Perhaps the sign? $\endgroup$ – Willie Wong Sep 27 '12 at 7:46
  • $\begingroup$ As for the sign, there is a minus sign from the rule and the derivative is negative, I think -*- is positive. $\endgroup$ – Daniel Lårs Sep 27 '12 at 7:48
  • $\begingroup$ Ah, sorry, I wrote my comment before I saw your response to yohBS. But in the end you are missing also a factor of $m-d-\mu$ in addition to the $\sigma^{-2}$. $\endgroup$ – Willie Wong Sep 27 '12 at 7:53

Let $F(y) = \int_y^\infty x f(x)~ \mathrm{d}x$, we can write it also as

$$ F(y) = - \int_{\infty}^y x f(x)~ \mathrm{d}x$$

so by the fundamental theorem of calculus

$$ \frac{\mathrm{d}}{\mathrm{d}y} F(y) = - y f(y) $$

Let $G(\sigma) = F( (m-d-\mu)/\sigma )$. Then we have that by the Chain rule (not the Leibniz rule!) that

$$ \frac{\mathrm{d}}{\mathrm{d}\sigma} G(\sigma) = \frac{\mathrm{d}}{\mathrm{d}\sigma} F\left( \frac{m - d - \mu}{\sigma}\right) = F'\left( \frac{m - d - \mu}{\sigma}\right) \cdot \frac{\mathrm{d}}{\mathrm{d}\sigma} \frac{m-d-\mu}{\sigma} $$

and you can take it from here. :-)

  • $\begingroup$ ♦ thank you a lot. You suggested me the simplest way. $\endgroup$ – Daniel Lårs Sep 27 '12 at 7:54
  • $\begingroup$ I want to be certain about the result, would the final result look like this?: $(((m-d-μ))/(σ²))(((m-d-μ))/σ)f(((m-d-μ)/σ))$ ? $\endgroup$ – Daniel Lårs Sep 27 '12 at 8:34
  • $\begingroup$ That looks okay to me, provided all the parentheses are correctly closed (I didn't check that). $\endgroup$ – Willie Wong Sep 27 '12 at 10:32

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