Normalising Constant

Meaning and Explanation

The normalising constant is a crucial concept in probability theory and statistics, particularly when dealing with probability density functions (PDFs). It ensures that the total probability of all possible outcomes sums to one, making the PDF valid.

Meaning

In a continuous probability distribution, the normalising constant adjusts the function so that the area under the curve (the integral of the PDF over its entire range) equals one. This is essential because probabilities must sum to one by definition.

For example, consider a function $f(x)$ that we want to use as a probability density function. To make $f(x)$ a valid PDF, we need to find a constant $C$ such that:

$\int_{-\infty}^{\infty} C \cdot f(x) , dx = 1$

Here, $C$ is the normalising constant. Once $C$ is determined, the function $C \cdot f(x)$becomes a valid PDF.

Calculation Method

The normalising constant is typically calculated by integrating the unnormalised function over its entire range and then taking the reciprocal of the integral. This process ensures that the integral of the normalised function equals one.

Example Calculation

Let's consider a simple example where $f(x) = e^{-x^2}$. This function is not normalised because its integral over all $x$ does not equal one.

Step 1: Calculate the Integral of $f(x)$

We first need to calculate the integral of $f(x)$ over all $x$:

$I = \int_{-\infty}^{\infty} e^{-x^2} , dx$

This integral is known as the Gaussian integral, and its value is $\sqrt{\pi}$.

Step 2: Calculate the Normalising Constant

The normalising constant $C$ is the reciprocal of the integral:

$C = \frac{1}{\sqrt{\pi}}$

Step 3: Normalise the Function

The normalised probability density function $g(x)$ is:

$g(x) = \frac{1}{\sqrt{\pi}} e^{-x^2}$

Now, the integral of $g(x)$ over all $x$ equals one, making $g(x)$ a valid PDF.