Talk:Probability mass function

Including a Citation

I was about to "correct" the half-closed interval in the first paragraph. Is there a good way to link this to article on Intervals? --Adolphe Youssef (talk) 18:10, 13 March 2008 (UTC)

I added a wikilink to handle this. Melcombe (talk) 16:48, 4 August 2008 (UTC)

Specific formulation as a function?

What is the definition of the funcrion to be? At present it is in the form f(x) where x is essentially continuous. The reference I added use the form f(j) where j is an integer which indexes the support points of the distribution. Possibly we need both. Melcombe (talk) 16:51, 4 August 2008 (UTC)

PMF may fail to be diffable even when it vanishes

Let f_X be defined as: f_X(x) = x/2, for x = 1, 1/2, 1/4, 1/8, 1/16, ...; and f(x) = 0 otherwise. Then X never takes on the value 0, yet f_X is not differentiable at 0. 207.62.177.227 (talk) 18:19, 20 November 2008 (UTC)

Open Interval

Why not just use a closed interval for pdf?

My guess is that the author was either thinking of (a) expressing the integral in terms of the CDF: P(a < X ≤ b) = F_X(b) − F_X(a); or (b) the fact the semi-infinite-closed intervals (− ∞, x] are often used to generate the Borel sigma-algebra on the reals. 75.95.125.245 (talk) 05:47, 24 November 2008 (UTC)

Definition

Am I the only person who finds the definition unhelpful? The introduction of the function X is not well-explained, and doesn't match most people's naive understanding, in which a pmf simply maps each element in the sample space S to a probability. Instead of this:

$f_{X}(x)=\Pr(X=x)=\Pr(\{s\in S:X(s)=x\}).$

it seems to me more natural to write:

$f(s)=\Pr(X=s)$ for each $s\in S.$

If we write this, it's then easy to observe that if there is a natural order on S we can introduce the cmf, and give "tossing a coin" and "rolling a die" as examples of cases where such an order isn't/is present.

I think the problem I have with the function X is that it seems somehow redundant: it seems to map each element of the sample space to a number, which in turn is then mapped to a probability. X doesn't seem to add anything to the definition. Or have I missed the point?

RomanSpa (talk) 21:42, 16 December 2013 (UTC)

Looking at the revision history of the definition, the introduction of the function X seems to originate in an earlier definition in which the sample space S is a subset of the real numbers R, which strikes me as a rather narrow way of thinking about sample spaces - it would eliminate the possibility of having {heads, tails} as the sample space from tossing a coin, for one thing. What do other people think? RomanSpa (talk) 22:02, 16 December 2013 (UTC)

I totally support your confusion. I do not understand why nobody else, no professional probabilists, could mention the overcomplication or reason for it either. I even asked this in stackoverflow. It seems that your guess, to be able to map heads/tails is not totally satsifactory because you could map heads/tails directly to probabilities. My question also disputes the discrimination between arbitrary samples and real outcomes. The article says that random variable maps samples to outcomes. Yet, i read everywhere that outcomes is just another name for samples. The real images of random variable are called output events: "random variable is a measurable function X that maps the measurable space (Ω, F ) to another measurable space, usually the Borel σ-algebra of the real numbers (R, R). We can see that the formal definition is saying the same thing as the basic definition. Consider an output event A ∈ R." Random variable calls them simply outputs: "The formal mathematical treatment of random variables is a topic in probability theory. In that context, a random variable is understood as a function defined on a sample space whose outputs are numerical values. A random variable is a real-valued function defined on a set of possible outcomes, the sample space Ω. That is, the random variable is a function that maps from its domain, the sample space Ω, to its range, the real numbers or a subset of the real numbers. It is typically some kind of a property or measurement on the random outcome (for example, if the random outcome is a randomly chosen person, the random variable might be the person's height, or number of children)." Might be it is a new approach (I have studied like you, that random variables just takes this or that random value with some probability) but we should ask "why there is a mapping Ω -> R, what is the point of that?" in the random variable. It says that boolean, vector, process and other extensions of random variable are admissible under term random element. "These more general concepts are particularly useful in fields such as computer science and natural language processing where many of the basic elements of analysis are non-numerical. Reduction to numerical values is not essential for dealing with random elements: a randomly selected individual remains an individual, not a number." You may need the real value to compute the expected value. However, I do not understand what is left to the random variable if it does not map samples to reals and what this mapping has to do with probability mass function. You should be able to define the heads and tails probability regardless of their numeric value, which may not exist at all. Why should I define one? --Javalenok (talk) 22:20, 16 July 2014 (UTC)

Equation for x success in n trials?

The Binomial distribution article claims that the Probability mass function is an equation found in most Statistics textbooks (the product of n choose k, p^k, and q^n-k). However, Probability mass function doesn't include this. I get that this equation is the result of what appears to be Probability mass function's rigorous mathematical description (I'm not qualified to verify it), but is there a way to connect these articles cleanly? In other words, someone reading Binomial distribution and clicking on the link to Probability mass function will probably find themselves completely lost. Baltakatei 23:32, 12 August 2019 (UTC)

A probability mass function is a kind of function, not a specific function. What the article on the binomial distribution says (quite correctly) is that the probability mass function of a binomially distributed variable is, as you say, the product of n choose k, p^k, and q^n-k, for k successes in n trials with a probability of success p. -Bryanrutherford0 (talk) 14:25, 13 August 2019 (UTC)

A different formal definition for PMF in Probability space#Discrete case

This is the definition there: "..by the probability mass function $p:\Omega \to [0,1]$.." and in here it's: "..It is the function p: $\mathbb {\mathbb {R} }$ $\rightarrow$ [0,1]..", what is the right definition than? — Preceding unsigned comment added by Shalevku (talk • contribs) 11:47, 2 March 2020 (UTC)

Those definitions are essentially interchangeable; one uses Omega to refer to the sample space of the experiment, and the other uses Heavy R to refer to the set of real numbers. So, one is thinking of the possible outcomes of an experiment in more general terms, while the other is thinking of them more specifically as numbers. All the other elements of the two are the same. -Bryan Rutherford (talk) 13:44, 2 March 2020 (UTC)