I do not understand the utility of calculating the density probability in a point in the case of a continuous variable (for example in the case of the normal distribution). Some CASIO calculator include a function ("normal P. D.") which calculate the value of the density function of normal at a point. What is the usefulness of this calculation? Why these calculators include this function? Thanks.

At a point like, continuous age distribution and an age of 21years, what is the probability of an age of 21 years in sample, or do you mean probability of an age of 21 years or younger in a sample.

For most calculation purposes the CDF is more important than the PDF. That isn't to say the PDF isn't useful though. On a calculator one might want to use the PDF for plotting purposes so you can make a graph of what the distribution actually looks like. Other uses might include random number generation (there are some methods that require the PDF of a distribution to generate from) or possibly to generate the CDF 'by hand'. If one wanted to calculate the likelihood for some parameters for normally distributed data you would also need the PDF as well.

The problem is that you can read at the CASIO manual (fx-9860GII): "Normal probability density calculates the probability density of a normal from a specified x value". I can understand this but at the Spanish translation of this manual (fx-9860GII) you can read (I have made a translation into English): "The normal probability density calculates the probability of a process governed by this distribution for a given value of x" (original text that appears in the Spanish manual -> "La densidad de probabilidad normal permite calcular la probabilidad de un proceso regido por dicha distribución para un valor de x determinado"). I think this is a mistake: the term "probability density" is not the same that the term "probability". In a continuous variable, the probability at a point is always zero and any student that can read the text in the Spanish manual can understand that the function that incorporates Casio calculator calculates the probability at a point in the case of a normal distribution which is not correct. I do not know if you can understand me (English is not my native language). I appreciate yours time for listening my question. Thanks a lot.