askvity

What is the Derivative of the Probability Density Function?

Published in Probability Derivatives 3 mins read

The derivative of the probability density function (PDF) is not a standard defined concept in probability theory, as the PDF itself is defined as a derivative. Instead, the PDF, often denoted as f(x), is the derivative of the cumulative distribution function (CDF), F(x).

Understanding Probability Density Function (PDF) and Cumulative Distribution Function (CDF)

To clarify, let's delve deeper into the relationship between PDFs and CDFs:

  • Cumulative Distribution Function (CDF): The CDF, denoted by F(x), gives the probability that a random variable X takes a value less than or equal to x. Mathematically, it's expressed as F(x) = P(X ≤ x).
  • Probability Density Function (PDF): For continuous random variables, the PDF, f(x), gives the probability density at a specific value x. It is not the probability itself but provides relative likelihoods.

The fundamental relationship is that the PDF is the derivative of the CDF:

   f(x) = d/dx F(x)

This is the very definition of a PDF as stated in the provided reference: "The probability density function (pdf) f(x) of a continuous random variable X is defined as the derivative of the cdf F(x): f(x)=ddxF(x)."

Implications

Since f(x) is itself a derivative, taking its derivative (f'(x) or d/dx f(x)) results in the second derivative of the CDF (F''(x)). This second derivative doesn't have a direct, commonly used name or a direct probabilistic interpretation. It is not standard practice to define a special purpose for the derivative of the PDF.

Example

Let's assume a simple example where the CDF, F(x), is given by:

  • F(x) = 1 - e-λx for x ≥ 0, where λ is a constant.

Then:

  1. Find the PDF:
    • f(x) = d/dx (1 - e-λx) = λe-λx
  2. Find the derivative of the PDF:
    • f'(x) = d/dx (λe-λx) = -λ2e-λx

Here, f'(x) is the derivative of f(x), the PDF. Notice, that this derivative does not have a specific interpretation relating to probability.

Key Takeaways

  • The PDF is defined as the derivative of the CDF.
  • The derivative of a PDF is the second derivative of the CDF.
  • There is no commonly used standard name or direct probabilistic interpretation of the derivative of a PDF. It's mostly used in further analysis and it’s not a commonly used concept.

Related Articles