Probability Density Function

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 any one 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.

Probability Density Function

Definition

The Probability Density Function(PDF) is the probability function which is represented for the density of a continuous random variable lying between a certain range of values. It is also called a probability distribution function or just a probability function. However, in many other sources, this function is stated as the function over a general set of values or sometimes it is referred to as cumulative distribution function or sometimes as probability mass function(PMF). But the actual truth is PDF is defined for continuous random variables whereas PMF is defined for discrete random variables.

The probability density function is defined in the form of an integral of the density of the variable density over a given range. It is denoted by f (x).  This function is positive or non-negative at any point of the graph and the integral of PDF over the entire space is always equal to one.

Probability Density Function Formula

In case of a continuous random variable, the probability taken by X on some given value x is always 0. In this case, if we find P(X = x), it does not work. Instead of this, we require to calculate the probability of X lying in an interval (a, b). Now, we have to calculate it for P(a< X< b). This can be done by using a PDF. The Probability distribution function formula is defined as,

\(P(a< X< b)=\int_{a}^{b}f(x)\)

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), then the pdf is given by \(P(x)=\int_{a}^{b}f(x)dx\)
  • 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, i.e. \(\int_{-\infty }^{\infty }f(x)dx=1\)
  • 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.

Probability Density Function Example

Question:
The pdf of a distribution is given as \(f(x)= \left\{\begin{matrix}x; for\ 0< x< 1 \\ 2-x;for \ 1< x< 2 \\ 0;for\ x> 2 \end{matrix}\right.\).

Calculate the density within the interval (0.5< x< 1.5)

Solution:
P(0.5< x< 1.5)=\(\int_{0.5}^{1.5}f(x)dx\)

=\(\int_{0.5}^{1}f(x)dx+\int_{1}^{1.5}f(x)dx\)

=\(\int_{0.5}^{1}xdx+\int_{1}^{1.5}(2-x)dx\)

=\(\left ( \frac{x^{2}}{2} \right )_{0.5}^{1}+\left ( (2x-\frac{x^{2}}{2}) \right )_{1}^{1.5}\)

=3/4

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A perpendicular bisector of a given line segment

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