What is the difference between a distribution function, a probability function and a density function?

Answer

Dear Jan,

suppose a random variable X that can take on negative and positive values. Then X can always be characterized by its distribution function (also called cumulative distribution function):
F_X(x)=Prob[X≤x]so the probability that the random variable is smaller than x, with -∞X(x) increases only at discrete points). Because of the difference between these two special types, they can also be characterized in a separate way, namely discrete random variables by the probability function and continuous random variables by the density function. These are sometimes more intuitive to us than the more general characterization of the distribution function. We’ll take a closer look at both.

Thus, discrete random variables have a countable number of possible values. To settle the mind, let us take the whole numbers. Then you can very easily characterize the random variable by the probability function (also called mass function), given by
p_X(n)=Prob[X=n]so the probability that X is equal to n, for all n integers.
Indeed, only the integer values ​​can occur with a non-zero probability, so the probability function characterizes the random variable as well as the distribution function.

You cannot do this for continuous random variables, since the probability of a certain value occurring is zero (there is a continuum of possibilities, so the probability of that one number occurring is zero). What you can do to characterize the continuous random variable in a number x is to express what the probability is that the variable lies in a small interval with x as a starting point. This is what the density function (also called density) expresses:
f_X(x)=Prob[x<X≤x+dx]/dx, with dx infinitesimally small. This therefore expresses the probability that X lies in an infinitesimally small interval starting from x, and this relative to the length of the infinitesimally small interval. It turns out that f_X(x) is nothing but the derivative of F_X(x) into x.

In summary: every random variable can be characterized by its distribution function, but for strictly continuous and discrete random variables there are alternative characterizations, viz. the probability function and the density function.

Kind regards,
George Walraevens