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When is c.d.f $F_X(s)=P(X<s)=P(X \le s)$?

Particularly, why/when is the last equality true, since usually the c.d.f. is defined as

$$F_X(s)=P(X \le s)$$

but I've seen application of the c.d.f. as:

$$F_X(s)=P(X<s)$$

mavavilj
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    Surely you see that the event $X\le s$ is $(X<s) \cup (X=s)$? Then apply basic probability rules about mutually exclusive events. Now when is the probability of the second component (i.e. $X=s$) non-zero? So when is it 0? – Glen_b Feb 28 '17 at 13:41

1 Answers1

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For a continuous CDF, there won't be a difference since a single point has measure zero (i.e. $P(X=x) = 0$ for continuous RVs). See here:

https://en.wikipedia.org/wiki/Cumulative_distribution_function

Edit: clarified some ambiguity pointed out by a commenter

Tim
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Matt
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    This is not a matter of convention: for discrete rv's, $P(X<s)\ne P(X\le s)$ when $P(X=s)>0$. – Xi'an Feb 28 '17 at 13:59
  • @Xi'an Ah, I see where my comment was unclear. I will fix it. My apologies to OP. – Matt Feb 28 '17 at 14:37