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i asked a similar question yesterday , i think my way was not proper ,i received vague and confusing answers, so i asked it again clearly specifying what i actually want to ask

This is what i found about symmetric distributions, written in notes that i have : The distribution of rv (random variable) $X$ is symmetric about $a$ if $$ P ( x \le a - x ) =P ( x \ge a + x ) \qquad \forall x \in \mathbb{R} $$ I just want to confirm ,.,is this " if " correct ? i think it must be " iff " i mean it is clear to me that if $x$ is symmetric about $a$ it will satisfy above equation, but the converse must also true ? isn't it ?

ANUJ NAIN
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  • You already asked this Q: http://stats.stackexchange.com/questions/264087/definition-of-symmetric-random-variable-in-terms-of-distribution-function/264107?noredirect=1#comment505369_264107 what is wrong with the answers there? – kjetil b halvorsen Feb 26 '17 at 15:42
  • i asked a similar question yesterday , i think my way was not proper ,i received vague and confusing answers, so i asked it again clearly specifying what i actually want to ask – ANUJ NAIN Feb 26 '17 at 16:01
  • What you have written in your notes is the **definition** of what is meant when we say that $X$ has a symmetric distribution about $a$. The use of "if" is perfectly standard because everyone understands that definitions are "if and only if" statements and that the "if" in the definition is to be interpreted as "if and only if". So, the usage of "if" is correct in common usage, but if you insist that it must be written as "If and only if" or "iff" so that everybody understands that you too understand the definition correctly, that's fine by us. In short, "Yes, the converse is indeed true". – Dilip Sarwate Feb 26 '17 at 16:58

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