In order to talk about logical connectives in probability you should develop an appropriate formal language. Then you have all connectives. See e.g my article "The Metamathematics of Probability, in my RG page.

I am surprised that historical data tells you that the temperature in a city has *exactly* a triangular distribution, and *exactly* this particular triangular distribution.

Thank you all for your replies.

Richard

Yes, you have right. The triangular distribution would be strange, but it is only a simple example. I have not researched the temperature distribution in reality. I would like to know, does negation of variable exist in statistic? In example, I gave an easy pdf example.

For example, in fuzzy sets theory, if we have the fuzzy number: short, we can indicate easily: not short. If we have pdf of X, can we find pdf of ~X?

You asked "What is negation of the typical temperature for city Y in January". Your graph shows the actual historical distribution of temperatures in the city in January. Before asking about the negation, answer the question: what is the typical temperature? Before answering that question we have to know what you mean by "typical".

Seems to me that "the typical temperature" is a fuzzy notion; so "not the typical temperature" is fuzzy too. Neither has a probability distribution. Neither is a random variable.

There is no pdf of "the typical temperature" (it isn't a random variable). There is no pdf of "not the typical temperature". For instance, the temperature of the surface of the sun is not the typical temperature of your city in January.

"Not the typical temperature" is doubly fuzzy. Are you talking about the temperature of something? And if so, of what? Are you talking about temperatures which do occasionally occur in the city in January, or also other temperatures?

Yes it's my mistake in the description. Let's say that it is the pdf of air temperature for city Y in January (not typical). The air temperature is measured in every day. If we take into account last 100 years (for example), then we get pdf of air temperature (now it is random variable). Now, I see that negation of the air temperature does not make any sense, because the negation would be defined asprobabilty of unlisted air temperatures.

As summary, a random variable has not negation. Do you agree with this sentence?

Within conventional probability theory, there is no "negation of a random variable".

Moreover, I do not see any need to invent such a notion. But then I am not a fan of "fuzzy [whatever]". IMHO the world would be a better place without it.

Well perhaps there is a bit of confusion. Out of your observations you can build a histogram with, of course, negative and positive temperatures. This has nothing to to with a negative probability which is represented on the y.axis. After this you can try to use a model of "famous" pdfs and estimate their parameters by using ML, or if Multimodal EM with a mixed PDF composition for further analytic studies. Yet there is trouble around the corner, when you have negative observations and want to use for example the gamma-pdf which only "allows" positive observations. Im not sure about using a offset on the data, then fitting and subtracting the offset afterwards (Making sure the area under the curve stays 1 all the time of course).

My interpretation of your question is: given the pdf of a random variable X, what is the pdf of (-X)? If my interpretation is correct, then there is a very simple answer to your problem. Write Y=(-X), and its pdf as f_Y(y). Similarly, X has pdf f_X(x). Then use the theory on functions of random variables (e.g. Rice, "Mathematical Statistics and Data Analysis" section 2.3 (2nd ed. 1995)) to find that f_Y(y) = f_X(x=-y).

Thus, not very surprisingly, the pdf of Y for Y=-4 equals the pdf of X for X=+4 etc.

I think that you mean negation more in the Boolean sense, and not arithmetic negation. I agree with others than you cannot simply define a random variable directly from ~T. But, perhaps there are other ways to accomplish what you want to do. Say that the temperature T of a city is a random variable of interest with pdf f(T). You could define a new random variable A = T - M, where A is usually called the "anomaly" and M is a measure of central tendency (e.g., mean or median of T). You can estimate the pdf of A with real data by building a histogram of the difference between measurements of T and M. Consider M is the median of T. So, if some measurement of T, say T1, is below the median temperature, then you could calculate p = 1/2 - P( A < T1 - median T ). From that you could form the statement: The probability that temperatures are not colder than the median temperature than T1 is p. Does that get you closer to what you want to do?

There is a sense in which random variables may be obfuscating to a still defensible interpretation of the question, i.e. Boolean negation in Kevin's words.

Imagine we are all Bayesians and believe that, given whatever information, there exists a probability distribution describing exactly the uncertainty we feel about any unknown value. The unknown value is the temperature at the city (let's assume we have agreed on what exactly is the temperature we are talking about, e.g. so that day-night differences are out of the picture).

That value T is not a random variable, but still it makes sense to speak of the probability distribution of T conditional on the information I have being `T is typical' or, on the contrary, `T is not typical'. Then the question can be re-read as: Given the distribution of T | `T is typical' (which might be or not be the historical distribution, but it wouldn't be a non-sensical choice), how to determine the distribution of T |`T is not typical' ?

Then the question is whether the `negated' distribution is a function of the original distribution or not, or whether we feel it should be or needn't be. In the Bayesian modelling, obviously P(A | not B) is not a function of P(A | B) alone. In the fuzzy modelling, which according to Richard the world might be better off without (insert sad smiley here), `T is not typical' is modelled in a way which is a function of `T is typical', using a truth-functional logical approach as opposed to the non-truth-functionality of the probability approach.

I hope this reply still has something to do which Wojciech's original concerns :)

Use transformation of random variable:  take transformation of Z  = -X and find the distribution