Comments about "Probability density function" in Wikipedia
This document contains comments about the article Probability density function in Wikipedia
 The text in italics is copied from that url
 Immediate followed by some comments
In the last paragraph I explain my own opinion.
Contents
Reflection
Introduction
The article starts with the following sentence.

In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would be close to that sample.

Okay.

In other words, while the absolute likelihood for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample.

Okay and not okay. The above two sentences are rather complicated. The major problem is that there exist a difference between statistics, which implies data gathering and chance or probability which implies a certain type of mathematics.










1 Example

Suppose bacteria of a certain species typically live 4 to 6 hours.

This fact has to be established by means of 1000 experiments.

The probability that a bacterium lives exactly 5 hours is equal to zero.

That depents how the lives of each bacterium is measured. In hours? in minutes ? or in seconds?
This comment seems strange but it is important

A lot of bacteria live for approximately 5 hours, but there is no chance that any given bacterium dies at exactly 5.00... hours.

Again that depents how the lives of bacteria are measured.

However, the probability that the bacterium dies between 5 hours and 5.01 hours is quantifiable.

The answer requires the definition of probability


2 Absolutely continuous univariate distributions










3 Formal definition










3.1 Discussion










4 Further details










5 Link between discrete and continuous distributions










6 Families of densities










7 Densities associated with multiple variables










7.1 Marginal densities










7.2 Independence










7.3 Corollary










7.4 Example










8 Function of random variables and change of variables in the probability density function










8.1 Scalar to scalar










8.2 Vector to vector










8.3 Vector to scalar










9 Sums of independent random variables










10 Products and quotients of independent random variables










10.1 Example: Quotient distribution










10.2 Example: Quotient of two standard normals










11. See also
Following is a list with "Comments in Wikipedia" about related subjects
Reflection 1  Statistics.
If you want to understand the concept "probability density function" you have to start with an experiment.
 Suppose you want to study bacteria, specific how old they become. That means you have monitor individual bacteria, To do that you need a stop watch (I expect many) The stop watch is started when the bacteria
is "born"and is stopped when the bacteria is considered "dead". The time in seconds is written down in your logbook. The same is done for 1000 occurences. This finishes step one.
 The next step is, by going through the data in your log book, to find the age of the bacteria that lived the shortest. That will be 4 years or 48 month.
At the same time you should find the oldest. That will be 6 years or 72 month.
What that means that the actual age of each bacteria will be somewhere in between 48 and 72 month i.e. a period of 24 months.
Next you create 24 containers. Each container will get a year indication and a month indication.
 The next step is to start with entry 1 in the logbook and put a token in the container that shows the final age of the bacteria #1. This is done for all the 1000 entries.
When you are finished the number the of tokens in each container shows the number of bacteria that die in a certain month. The highest number should be in month 12 and 13.
 To get the probability density function you should divide the number in each container by 1000.
Reflection 2
Reflection 3
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Created: 20 January 2022
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