Probability Density Function | Research & Encyclopedia Articles

Probability Density Function

The probability density function, often denoted as pdf, is a function that describes the probability of finding a random variable within a defined range. The significance of the pdf, f(x), is that f(x)dx is the probability that the random variable x' is in the interval (x, x + dx). This is often written as P(x x' x + dx) = f(x)dx. Since f(x)dx is a probability it is unitless and therefore f(x) has the units of inverse random variable units. It is also possible to define the probability of finding the random variable somewhere in a finite interval. P(a x b) = ^b_a f(x)dx, where the finite interval is [a, b]. This probability is equal to the area under the curve defined by f(x) from a to b.

As with other probability distributions the pdf has some restrictions. Since f(x) is a probability density it must be positive for all values of the random variable x: f(x) 0, - < x < . The probability of finding the random variable somewhere on the real axis must be unity, that is it must be possible to find the random variable on the axis somewhere: _- f(x')dx' = 1. These are the only two restrictions that must be satisfied for a function to be a probability density function. The expectation of the probability density function is defined as E(x') = _- xf(x)dx = . This integral must be convergent for this to be true.

There are three common, important probability density functions: the normal probability density function, the exponential probability density function, and the Cauchy probability density function. The normal probability density function is important because the normal random variable is a frequently used model in statistical theory. This distribution is often called the bell curve or Gaussian distribution and its probability density function has the form f(x) = (exp(-1/2x²))/(2). This probability density function has a mean of 0 and a standard deviation of 1. The exponential probability density function arises in the study of the Poisson distribution. It has the form f(x) = a exp(-ax) (0 x < ) where a is positive for x 0. The Cauchy probability density function, or as it is sometimes called the Lorentzian distribution, describes a resonance behavior and is usually written as f(x) = 1/[((1 + x²)](- < x < ). It does not have a mean value or a variance since the integral does not converge. As a result of this it has some unique properties such that if independent random variables have a Cauchy distribution then the average of those variables also has a Cauchy distribution. The variability of the average is identical to that of a single observation.

Probability density functions play an important role in statistical analysis of events. These functions are useful in determining the likely-hood of finding a random variable in a specified interval. The probability density function is related to another important idea, the cumulative distribution function, as f(x) = F'(x), where F(x) is the cumulative distribution function. This relationship is important in many engineering and mathematics applications.