Probability Density Function (PDF)

Probability Density Function (PDF)

The cumulative distribution function (CDF) can give useful information about discrete as well as continuous random variables. However, the probability density function (PDF) is a more convenient way of describing a continuous random variable. The probability density function f_X(x) is defined as the derivative of the cumulative distribution function. Thus, we have,

Properties of PDF

Property 1: The CDF can be derived from PDF by integrating it i.e.,

Proof :

According to the definition of PDF , we have

Integrating both the sides, we get

It is important to note the upper limit of integration. It is not + ∞ but, it is ‘x’. This is because F_X(x) has been defined as the probability of X ≤ x.

Thus,

But, F_X(– ∞) = 0

Therefore,

or,

Property 2 : PDF is a non-negative function for all values of x i.e.,

f_X(x) ≥ 0      for all x

Reasoning: As we know that CDF is a monotone increasing function. PDF is the derivative of CDF and the derivative of a monotone increasing function will always be positive.

Property 3: The area under PDF curve is always equal to unity.

Therefore,

Proof :

As per the definition of PDF, we have

Integrating both sides, we get

But,

Therefore,