Is there any general formulation procedure of probability density functions? [closed]

There are many probability density functions for continuous variables around the world. Unlike the probability mass functions of discrete variables, these PDFs do not directly give you the probability. You have to integrate over a range to get a probability from a PDF and the area under the PDF must always be 1. It's understandable that density and relative frequency are quite different. Usually density is found by dividing the relative frequency by the range of the group. For example,

Group       Frequency       Relative frequency       Density
10-20          30                 0.30                0.03
20-30          20                 0.20                0.02
30-40          50                 0.50                0.05

So, if you draw the density against the range of values then the area under this graph is 1.

My question: is there any common procedure through which the PDFs of different continuous variables have been derived? For example, how did the PDF of Normal or Exponential or Maxwell distribution come about? Probably the PDFs have been normalized so that the area under the curve becomes 1. Could someone explain this please? Different density functions have different mathematical forms. I want to know the rules of how they came to be.

closed as too broad by whuber♦Apr 26 at 13:48

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• Generally, PDF is a function that outputs non-negative values that integrate to 1. So the procedure is simple: find a function that has this property. – Tim Apr 9 '15 at 12:27
• How to find that function? What mathematical methods are used? – Blain Waan Apr 10 '15 at 8:42
• Is this question too broad or is the answer simply "no, there is no such method"? (I have no idea, I am just suggesting that this might be the case) – Peter Flom Apr 27 at 11:19
• I have voted to reopen because I think the answer shows how your question is indeed answerable in a sense similar to Peter Flom alludes to. This bit of the question that asks "How did the PDF of Normal or Exponential or Maxwell distribution come? ... Different density functions have different mathematical forms. I want to know how they came" is probably too broad and ought to be removed. Clearly we cannot describe how every PDF was derived. ... – Silverfish Apr 27 at 11:49
• ... However, it is worth pointing out that if we want a random variables that has certain properties, there are ways to build on Carl's answer by finding a valid PDF that has the additional properties you want. For instance for the exponential distribution, you can find a valid PDF which has the "memoryless" property – Silverfish Apr 27 at 11:52

One follows the following rules:

1) Density functions are more general than probability density functions. A probability density function is one that has units of probability per unit measure, for example probability (of dying) per year (of life). A density function without probability, could be the relative concentration (of a drug in blood plasma) per hour, where the area under the time curve is a unit dose of drug.

2) Density functions have no negative dependent variable values. One cannot have, for example, negative relative concentration, or negative probability.

3) Density functions have what is called support. That is, they are defined on a range of $$(-\infty,\infty)$$, e.g., the normal distribution, or on $$[0,\infty)$$, and zero elsewhere, like the gamma distribution, or on $$[b\geq 0,\infty)$$, and zero elsewhere, where b is the scale of the type I Pareto distribution.

4) The total area of a density function over its support is 1. To be strict about the definition, a density function should not describe a discrete variable, like a model for the outcomes of a coin toss. Such variables are called mass functions, and if they involve probability then probability mass functions (pmf).

5) Finally, the answer to your question. Given the rules above, to make a pdf one takes any function that does not violate them, integrates over the support, and divides the function by that integral.

For example, take $$2 x-x^2$$ over $$x=[0,2]$$. Then as $$\int_0^2 \left(2 x-x^2\right) \, dx=\frac{4}{3}$$, the density function is then $$\frac{3}{4} \left(2 x-x^2\right)$$. This is shown on the plot below, where the original quadratic is in blue and the density function is in orange. Next, we think up a name, parabolic distribution works as well as any. One final point, this is not the standard parabolic distribution. That latter would be a density function with support on $$[0,1]$$. Now as to the other question. Suppose our parabolic density function has a height at any point that is proportional to the probability of an outcome occurring within some small interval from $$x$$ to $$x+\Delta x$$. Then the area under the curve or some part of the curve would represent the probability of outcomes occurring in that "window". For example, if we wanted to know what the probability of having an outcome occur between 0 and 1 on our parabolic density function, it would have an area of 1/2, by symmetry, such that the probability of such an outcome would be 50%.

You could remove the question that reads Normal and Maxwell distributions and in that way reduce the perception of the question's being "too broad." It actually isn't too broad. In Maxwell's case, the motivation was the physical equations implied by ideal gas assumptions, e.g., see the Maxwell distribution link. It also follows the rules above. In the case of the normal distribution, its development was more empirical, in the sense that it fit observations.

There are lots of other important rules about density functions that would influence reasons for creating them. For example, the convolution of two density functions is a density function. Another, if two density functions have mean values, their convolution has a mean value that is the sum of those means.