# Probability

In any random experiment, a chance, or probability, always exists that a particular event will or will not occur. The probability that event A will occur is the ratio of the number of outcomes favorable to A to the total number of possible outcomes.

You can assign a number between zero and one to an event as an indication of the probability that the event will occur. If you are absolutely sure that the event will occur, its probability is 100% or one. If you are sure that the event will not occur, its probability is zero.

## Normal Distribution

The normal distribution is a continuous probability distribution. The functional form of the normal distribution is the normal density function. The following equation defines the normal density function f(x).

| (E)

The normal density function has a symmetric bell shape. The following parameters completely determine the shape and location of the normal density function:

- The center of the curve is the mean value = 0.
- The spread of the curve is the variance s2 = 1.

If a random variable has a normal distribution with a mean equal to zero and a variance equal to one, the random variable has a standard normal distribution.

## Probability Distribution and Density Functions

The equation below defines the probability distribution function F(x).

| (G)

where f(x) is the probability density function, , and

By performing differentiation, you can derive the following equation from Equation G.

| (H)

You can use a histogram to obtain a denormalized discrete representation of f(x). The following equation defines the discrete representation of f(x).

| (I)

The following equation yields the sum of the elements of the histogram.

| (J)

where m is the number of samples in the histogram and n is the number of samples in the input sequence representing the function.

Therefore, to obtain an estimate of F(x) and f(x), normalize the histogram by a factor of Δx = 1/n and let hj = xj.