# Probability Distribution Function for a Single Variable

One of the most important concepts in Finance is the concept of a random variable. For example this can be viewed as the outcome of throwing a die where the process is fixed by the outcome is not.

Form the definition of a random variable is derived the definition of the distribution function which is the probability that the final value of the random variable is less than equal to a given number. The other word for this term is cumulative distribution function.

Where $F(X)=p(X\leq x)$ is the cumulative distribution function

There are two possibilities in this case. When the variable takes discrete values and when it is continuous in nature. In case the variable is discrete in nature the distribution is obtained by summing the step values less than or equal to the reference number and the function that defines this variable is called the frequency function or the probability density function. When the variable is continuous in nature this is obtained by taking the integral of the probability density function over all possible values until the reference number. The probability density function is then the derivative of the first derivative of the distribution function with respect to the variable.

For the discrete case:

$F(X)= \sum\_{x(i)\leq x} f(x(i))$

For the continuous case:

$\int\_{-\infty }^{x}f(u)du$

In the continuous case the integral of the probability density function from negative infinity to complete infinity will be 1.