Probability Density Function

probability density function formula

Probability Density Function (PDF) is used to define the probability of the random variable coming within a distinct range of values, as objected to taking on anyone value. The probability density function is explained here in this article to clear the concepts of the students in terms of its definition, properties, formulas with the help of example questions. The function explains the probability density function of normal distribution and how mean and deviation exists. The standard normal distribution is used to create a database or statistics, which are often used in science to represent the real-valued variables, whose distribution are not known.

The PDF is the density of probability rather than the probability mass. The concept is very similar to mass density in physics: its unit is probability per unit length. To get a feeling for PDF, consider a continuous random variable X and define the function fx(x) as follows (wherever the limit exists):


                                                             f_{x}\left ( x \right )= \lim_{\Delta \rightarrow 0^{+}}\frac{P(x< X\leq x+\Delta )}{\Delta }

To determine the distribution of a discrete random variable we can either provide its PMF or CDF. For continuous random variables, the CDF is well-defined so we can provide the CDF. However, the PMF does not work for continuous random variables, because for a continuous random variable P(X=x)=0 for all x∈R.

The function fx(x) gives us the probability density at point x. It is the limit of the probability of the interval (x,x+Δ] divided by the length of the interval as the length of the interval goes to 0. Remember that

                                  P(x<X\leq x+\Delta )=F_{x}(x+\Delta )-F_{x}(x).

So, we conclude that

                                       f_{x}\left ( x \right ) = \lim_{\Delta \rightarrow 0^{+}}\frac{F_{x}(x+\Delta )-F_{x}(x)}{\Delta } = \frac{\partial F_{x}\left ( x \right )}{\partial x} = {F}'_{x}\left ( x \right )

we have the following definition for the PDF of continuous random variables:

A continuous random variable with an absolutely continuous CDF Fx(x). The function fx⁡(x) defined by

                                            f_{x}\left ( x \right )= \frac{\partial F_{x}\left ( x \right )}{\partial x} = {F}'_{x}\left ( x \right ) ,    if  Fx(x)  is differentiable by x

is called the probability density function (PDF) of X.

Probability Density Function Properties

Let x be the continuous random variable with density function f(x), the probability distribution function should satisfy the following conditions:

  • For a continuous random variable that takes some value between certain limits, say a and b, and is calculated by finding the area under its curve and the X-axis, within the lower limit (a) and upper limit (b)
  • The probability density function is non-negative for all the possible values, i.e. f(x)≥ 0, for all x
  • The area between the density curve and horizontal X-axis is equal to 1
  • Due to the property of continuous random variable, the density function curve is continuous for all over the given range which defines itself over a range of continuous values or the domain of the variable.

Applications of Probability Density Function

The following are the applications of the probability density function:

  • The probability density function is used in modelling the annual data of atmospheric NOx temporal concentration
  • It is used to model the diesel engine combustion
  • In Statistics, it is used to calculate the probabilities associated with the random variables.