Probability Density Function

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. The probability density function also called as probability distribution function. The probability density function is defined in the form of an integral of density of the variable density over given range. It is denoted by f (x).

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 probability density function.The probability Density function is defined by the formula,

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

Properties of Probability Density Function

Let x be the continuous random variable with density function f(x), it 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.
\(f(x)\geq 0\) , 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 variable.

Sample 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:
\(P(0.5< x< 1.5)=\int_{0.5}^{1.5}f(x)dx\) \(=\int_{0.5}^{1}f(x)dx+\int_{1}^{1.5}f(x)dx\) \(=\int_{0.5}^{1}xdx+\int_{1}^{1.5}(2-x)dx\) \(=\left ( \frac{x^{2}}{2} \right )_{0.5}^{1}+\left ( (2x-\frac{x^{2}}{2}) \right )_{1}^{1.5}\) \(=\frac{3}{4}\)

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Practise This Question

Which point is interior to ABC?