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I have these two definitions for "Probability distribution" and "Probability function":

Probability Distribution: Assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference. (Wikipedia)

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Probability Function: Assigns an event A a probability P(A) that represents the likelihood of event A occurring. (MITx 6.00.2x Introduction to Computational Thinking and Data Science)

Those are very similar yet slightly different definitions. Specifically, Probability Function seems to refer to a single event whereas Probability Distribution seems to refer to a collection of events. I've googled around but not found a satisfying answer as it seems that both terms are used in different contexts.

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  • $\begingroup$ There is a direct translation between the two statements afforded by the fact that an "event" is defined to be a measurable set. The use of "likelihood" in the second quotation is a poor choice, because in a technical sense likelihood is not the same as probability, even though it is used in that sense here. Evidently this is because the second quotation is attempting to be informal and "intuitive." That's fine, but such informal statements should not be too closely interpreted. $\endgroup$
    – whuber ♦
    Commented Nov 7, 2014 at 19:04
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    $\begingroup$ I wouldn't over-interpret the differences in those two. Formally, I wouldn't call either strictly correct (the first will lead to confounding of the meaning of the word 'distribution', for example, when I think it should make the idea distinct). On the other hand, wikipedia isn't - and shouldn't be - aimed at formal correctness, especially on more basic topics. $\endgroup$
    – Glen_b
    Commented Nov 8, 2014 at 0:18
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    $\begingroup$ I definitely think you should look into buying a dictionary of Statistics--you can find used copies on Amazon selling for cheap (you mainly just pay for shipping). I know the confusion that can arise when jumping from one online resource to another--a Stats dict. is great for concise, reliable, easy-to-understand definitions. Worth the investment. $\endgroup$
    – Steve S
    Commented Nov 30, 2014 at 6:09
  • $\begingroup$ Thank you Steve. I will peruse Amazon for a good statistics dictionary, and I'll see if the local used English bookstore has something as well. I did not even realize that there exist stats dictionaries. $\endgroup$
    – dotancohen
    Commented Nov 30, 2014 at 6:40

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My reading is that they are not the same thing. The following paragraph from the Wikipedia page for "probability density function" may help or may just confuse you further:

The terms "probability distribution function"[1] and "probability function"[2] have also sometimes been used to denote the probability density function. However, this use is not standard among probabilists and statisticians. In other sources, "probability distribution function" may be used when the probability distribution is defined as a function over general sets of values, or it may refer to the cumulative distribution function, or it may be a probability mass function rather than the density. Further confusion of terminology exists because density function has also been used for what is here called the "probability mass function."

Your definition of a "probability function" ("assigns a probability to an event") seems more applicable to a probability mass function, $f(x) = Pr(X=x)$, which is a function that gives the probability that a discrete random variable is exactly equal to some value. A probability density function, maps a continuous variable to the relative likelihood of that value as compared to other values. For given $x$, the value of the probability density function $f(x)$ may have a value greater than 1, which is a no-no when it comes to probability. For example, the uniform distribution on the interval [0, .5] has a value of 2 on that interval and 0 elsewhere. In order to obtain probability from the probability density function, one must integrate between the values of interest. This means, interestingly, that the probability of a given specific $x$ occurring is technically zero. This is because if in integrate $f(x)$ from $a$ to $b$ where $a=b$ you get $F(a)-F(a) = 0$ where $F(x)$ is the cumulative distribution function.

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    $\begingroup$ This analysis seems only tangential to the question, which clearly refers to probabilities rather than probability densities. The second paragraph is too limited: the definition of a probability distribution in the question is intended to apply to all distributions, whether discrete (as you assume) or not. $\endgroup$
    – whuber ♦
    Commented Nov 7, 2014 at 18:52
  • $\begingroup$ The question was about terminology, specifically the difference between "probability distribution" and "probability function". I am attempting to provide definition and distinction between the loose terms the questioner provided and the formal definitions of probability theory for "probability mass function" and "probability density function". $\endgroup$
    – Dalton Hance
    Commented Nov 7, 2014 at 19:33
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    $\begingroup$ Yes--but unfortunately because you seem to be writing about something different than the question asks, you answer appears to have little bearing on it. $\endgroup$
    – whuber ♦
    Commented Nov 7, 2014 at 19:34
  • $\begingroup$ Thank you. I deliberately did not mention the confusing term "Probability distribution function" as I was not sure where that fit in. I'm still not sure, but I'm at least becoming more familiar with the terms that I should know. $\endgroup$
    – dotancohen
    Commented Nov 7, 2014 at 20:19

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