An interesting difference: the probability density function of a continuous random variable is the derivative of the cumulative distribution function, as long as the derivative exists. Conversely, the cumulative distribution function of a continuous random variable is an improper integral of the probability density function.

An essential difference: the cumulative distribution function determines uniquely the random variable, but the are random variables who do not possess probability density functions. e.g., the discrete ones.

The probability distribution function (pdf) of a random variable X is a function of x such that (i) f(x) > 0 for every value x of X; (ii) the probability of an event A is P(A) = sigma ( f(x)) for every x in A. The (cumulative) distribution function of X is defined by

F(x) = P(X

Tow types of probability functiona discrete and continuouse

Usually, the distribution function means the cumulative distribution function (CDF, represented as F(x)) of a random variable (say X).

The probability distribution function shows the values of random variable X and the corresponding probabilities. For discrete random variable, the probability distribution function is named as probability mass function (pmf, represented as p(x))

and for the continuous random variable, it is referred to as probability density function (pdf, represented as f(x)).

For instance, if the random variable X is discrete, we can define the pmf as

p(x)=P(X=x)=x/6 ; x=1,2,3

and the CDF as F(x)=P(X

An interesting difference: the probability density function of a continuous random variable is the derivative of the cumulative distribution function, as long as the derivative exists. Conversely, the cumulative distribution function of a continuous random variable is an improper integral of the probability density function.

An essential difference: the cumulative distribution function determines uniquely the random variable, but the are random variables who do not possess probability density functions. e.g., the discrete ones.

If X is a continuous random variable (defined over a continuous sample space So), we define f(x), the probability density function (pdf) as the function satisfying the conditions:

1. f(x)>=0 (for all x€S, the sample space and

2. Integral of f(x)dx=1 (i.e. integration over the sample space S)

On the other hand;

Using the convention of the PDF being zero where it is not defined, we can define the distribution function F(x) by

F(x)=P(X

Distribution function is a test statistics or tool that can be used to test statistical hypothesis while probability distribution function is a distribution used to test the occurrence of an event at a given time and condition . Thank you

The distribution function essentially encompasses the distribution of the probability of both discrete and continuous functions. The latter being related to the probability distribution function.

Another difference as pointed out by George Stoica in a different form although) is that continuous probability distribution function tells you only the probability of a random variable being drawn from a unique interval, whereas in discrete ones it can be attributed to a specific discrete value

Distribution function could be probabilistic or nonprobabilistic. But the probability distribution function is the cumulative density functions of the probability density function or the probability mass function.