# Are “Probability Distribution” and “Probability Function” the same thing?

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)

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.

• 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. – whuber Nov 7 '14 at 19:04
• 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. – Glen_b Nov 8 '14 at 0:18
• 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. – Steve S Nov 30 '14 at 6:09
• 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. – dotancohen Nov 30 '14 at 6:40

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.