probability distribution function is acumulative of the probability distribution function Fx(a)= integral (f(x),x=-infinity to a)

Let f(x) be the probability density function and F(x) corresponding distribution function. Then Probability (a < X

@Subrata. what is the difference between them? are they same having different name?

They are not the same, since the distribution function F(X) measures cumulative probabilities P(X

Ana is correct they are not the same. I assume you are having a continuous r.v. X. In that case you can get back f(x) from F(x) as f(x)= D F(x). See any standard book you will get to know all these.

Probability density function is a function, describes the relative likelihood for this random variable to take on a given value.

Probability distribution function is defined as a function over general sets of values.

Probability density function is derivative of probability distribution function.

Probability density function by itself will not give you any probability and in fact probability of X taking a particular value is zero, for a continuous r.v. X. An interpretation is like this: f(x)dx gives you the probability that X belongs to a interval of (x-1/ dx , x+1/2 dx) of small length dx.

The probability density function f is a density or rate: probability per unit of X (the variable in question). For small positive h, the probability that X lies in an interval of length h around the point x is approximately f(x) times h.

The cumulative probability distribution function F is the cumulative probability of all values up to x. The probability that X is less than is equal to x is F(x).

Both density function (if there is one) and distribution function characterize "the probability distribution of X" which is, loosely speaking, the collection of all possible probability statements one might like to make about X. More precisely, all probabilities Prob(X lies in B) where B is any nice enough subset of the real line.

In probability and statistics it is important to distinguish between density, distribution, and distribution function. Three different things! However in physics, engineering, ... terminology is often sloppy and/or inconsistent, and definitely different from "official" mathematical terminology. So people in physics very very often talk about "distribution function" when a specialist in probability or statistics would say "density function". Too bad, that's the way it is. One needs to learn the languages and learn to translate. Eskimos have twenty different words for different kinds of snow but people who don't have to deal so intimately with snow day by day only have one or two.

Richard, I believe that we need to refine your definitions even more. When you say "The probability density function f is a density or rate: probability per unit of X (the variable in question). For small positive h, the probability that X lies in an interval of length h around the point x is approximately f(x) times h",

If we consider two close values of F, or Cumulative Distribution Functions (P; X(P)) we may obtain two small intervals: 1) h=X(Pi+1)-X(Pi) and 2) p=P(i+1)-P(i). Combining them we obtain a kind of density f= p/h, so when length interval h becomes very small and positive, f tends to the derivate dP/dX.

When we know F=X(P cummulative) we can derive dX/dP and its inverse is just the inverse p/h=dP/dX, that may be considered a kind of density of probability as a function of distributed variable X. So h length is approximately h=p* dX/dP.

I have observed that when we use very tiny p intervals the interval´s average found Xave tends to be somewhere in between the interval h -not necessarily centered-, and it is very close to the two limiting X values. And finally, the contribution of the interval to the total distributed mass may be expressed as dL=Xave*p for smalls intervals of P. In summary F and f are different things, but they are theoretically and structurally related to the same distribution.

I do not understand why the 19th. Century mathematics started the analysis from f, instead of starting from F to make explanations easier. Welcome to your comments, emilio