I understand that PDF is the derivative dXac/dK where Xac=cumulative population and K=variable value. So if we can express K as a continuous function of Xac, and we can estimate dK/dXac then PDF= 1/(dK/dXac). This is also related to histogram graphs. I have confirmed it using Pareto´s distributions for descending ordering. Is this vision sound to you? Is it tautological? What happens when dK/dXac=zero?
if dK/dXac = 0 it means that K is a constant which is not depending on Xac.
Dear Emilio
The probability density function (pdf), or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value. I think that a good explanation is on: http://en.wikipedia.org/wiki/Probability_Density_Function
You are interested in some "random variable" X (when teaching to my students, I use "random number" (r.n., suggested by an Italian guy: B. de Finetti)), which turns out to the proper name fot that. I use the language with which my students are familiar. For a discrete random number the probability natural question is:"what's the prob that X takes the value x*?". The answer is provided by the "probability function p" Prob(X = x*) = p*, a non-negative number. When we switch to continuos random numebrs (the ones having a distribution described by some density f(x), if we ask the same question as above "what's the prob that X takes the value x*?", the answer is deceiving: that prob is 0. The right question, for a continuous random number is: "what's the prob that X takes a value between x* and x*+h? with h -> 0".
The answer is:
P(x* < X < x+h) = f(x)h + o(h) with h -> 0
f isn't a prob, but it becomes (approximately) if multiplied by h.
You want a concrete example.
Well, what's your prob to die tomorrow? The question is subjectively dazzling, but crucial for instancefor the business of life insurance.
The so-called mortality rate m(x) = - l'(x)/l(x), being lx) the prob a guy, with age 0 will be alive at the age x. m(x) is haunted probability density function.
Well: m(x).h willa approx the prob you will die in 1 year: f(x).1 = f(x).
Some info about m(x) could improve our lifes..
m attains a min around x= 13, in line with our experience about adolescents. Then it increases exponentially with the age. For intermediate ages it's around 2%: be happy.
Best
Lorenzo
Thanks to Johaness, Guillermo and Lorenzo for your repplies and time. Lorenzo points out the analogy between PDFs (a limit when H tends to zero vs Variable) and Histogram (a concrete probability for a concrete variable interval), My observation is that histograms need to be ruled because each researcher builds his/her particular histogram for different relative frequences for the same data set. The second point is that if we have a good theoretical model of Lorenz curves as a referent, then we may check fittings, precision and simetry of a data sample, and construct CDF and histograms from small or big samples. I do not see the variable as aleatory, but instead as the result of some structural context over fractions of cummulative population -from zero to one after ordering them- that conform the sample and are selected in an aleatory process before ordering by size. Therefore the average of the cumulate population fractions tends so 1/2.
But my interpretations may be far from being good tips to the question asked. Thanks for feedback so far, emilio
For a function f(x), x defined in some [a, b), if the total integral is equal to 1, the variable is said to be random, and the function f(x) is called the 'density function' of x. For a random variable following some continuous probability law, this function is known as the 'probability density function'.
The original question was of interpretation.
What we can think as accepted is that, a random number X, admits a pdf f if its cdf F(x) = Pr (X
Since, it asked for the mathematical sense, we may say simply that a pdf is just an a.s. randon-Nykodim derivative with respect to lebesque measure in we are in the absolutely continuous case, and with respct to counting measure for the discrete case.
- Badges
- Science topic
- Similar topics
- Mathematics