Is it true that any two normal distributions with the same mean can be ordered w.r.t. the relation of a mean preserving spread? My intuition would be that this is true but I cannot come up with a formal proof.

Thanks in advance

  • $\begingroup$ Could you please clarify what you mean by a "mean preserving spread"? An example, perhaps? $\endgroup$ – Easymode44 Mar 20 at 16:18
  • $\begingroup$ A distribution F_B is a mean preserving spread of a distribution F_A if both distributions have the same mean and i if the area under F A from minus infinity to x is less than or equal to that under F_B from minus infinity to x for all real numbers x with strict inequality at some x. Intuitively, both distributions have the same mean but one distribution puts more probability weight on "the middle". $\endgroup$ – MMM Mar 20 at 16:22
  • $\begingroup$ I'm sorry, I thought it was a sui generis term. I should have researched before asking. $\endgroup$ – Easymode44 Mar 20 at 16:25
  • $\begingroup$ That's alright :) $\endgroup$ – MMM Mar 20 at 16:40

Hint: If both normal distributions have the same mean $\mu$, there is a positive constant $c$ such that $F_1(x + \mu) = F_2(c x + \mu)$ where $F_1$ and $F_2$ are the CDF's.

  • $\begingroup$ Thanks a lot, this does the trick. $\endgroup$ – MMM Mar 21 at 9:24

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