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.

$fx=\frac{dF(x)}{dx}$

If the cumulative distribution function is integrated over the complete range of values from -∞ to +∞ then it should be equal to 1.

$\int_{-\infty }^{+\infty }f(u)du=1$

Example of Density Functions (for the discrete case)

Take a case of rolling two dice at once. Since there are six faces on one side of the dice the total number of outcomes of the total of the addition of the two numbers that can appear on each face of the dice is 6*6=36. The range of final result is from 2-12 with varied frequencies representing the number of times each outcome can happen. The probability of each outcome is the frequency of each outcome divided by the total number of outcomes possible. The addition of all these probabilities should of course be 1. For any outcome the cumulative probability is the sum of the probabilities for each outcome lesser and equal to that outcome.

This is very useful in finance because the same approach can be applied to financial information like stock quotes, foreign exchange rates, commodity prices and exchange rates etc.

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