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In many different (serious and good) statistics books I find different definitions of CDF of a discrete RV. The difference is the equal sign at the index of the summation sign. The first is:

$$F(x) = P(X\leq x) = \sum_{x_i\leq x} p_i$$

whereas the second is:

$$F(x) = P(X < x)= \sum_{x_i< x} p_i$$

For most of my life I was convinced that CDF is a function returning the probability that a RV will take a value not greater than x, and therefore I am used to the first definition.

So which one is it?

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    $\begingroup$ I have never seen the second form in a text or reference book... can you please give a reference? $\endgroup$
    – jbowman
    Commented Mar 11, 2019 at 14:34
  • $\begingroup$ It's a matter of convention. In some disciplines (especially engineering-related ones), the CDF is defined as $\Pr(X \ge x),$ for instance. Thus there isn't going to be any "right" answer; the best we can hope for in an answer would be a survey of which disciplines tend to use which conventions. $\endgroup$
    – whuber ♦
    Commented Mar 11, 2019 at 15:10
  • $\begingroup$ Probability and statistics in examples page 52 $\endgroup$
    – Wojciech Artichowicz
    Commented Mar 11, 2019 at 15:18
  • $\begingroup$ Very interesting, @whuber. I'm in engineering and only CDF definition I've seen is $P(X\le x)$. I've seen CCDF as $P(X > x)$. I've scoured some common textbooks from various time periods. Mind if I ask what engineering fields? $\endgroup$
    – SecretAgentMan
    Commented Mar 11, 2019 at 16:19
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    $\begingroup$ @Secret Mathematically these distinctions are of no consequence. We could study distributions using any of the conventions for CDFs. After all, the CCDF of random variable $X$ is a simple transform of the CDF of $-X.$ Thus there aren't any conceptual issues raised. Perhaps the most important thing worth knowing is that when you're reading unfamiliar literature, confirm your understanding of the conventions. $\endgroup$
    – whuber ♦
    Commented Mar 11, 2019 at 17:24

1 Answer 1

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The former is correct. For continuous RV's the distinction is meaningless, i.e. $P(X\leq x)=P(X<x)$, since the probability of observing a single value of the RV is zero, since a single point on the real line has Lebesgue measure of zero. With discrete RV's the distinction is important because $P(X\leq x) \neq P(X<x)$.

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  • $\begingroup$ That is my opinion too. However, I can't understand how come such thing can be found in really decent books. What reference could I use to back that up? $\endgroup$
    – Wojciech Artichowicz
    Commented Mar 11, 2019 at 14:31
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    $\begingroup$ In which textbook did you see this? Casella and Berger will certainly have it correct. $\endgroup$
    – user240715
    Commented Mar 11, 2019 at 14:32

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