Processing math: 100%

Probability Density Function Calculator

Formula

To calculate the probability density function:

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

Definition

A Probability Density, often referred to in the context of Probability Density Function (PDF), is a statistical expression that defines a probability distribution for a continuous random variable as opposed to a discrete random variable. It describes the likelihood of a random variable taking on a specific value. The area under the curve of a PDF (between any two points) represents the probability that the variable falls within that range. The total area under the curve of a PDF is always equal to 1, representing the total probability of all possible outcomes.

Example Calculation

Let's assume the following values:

Point $x$ = 1
Mean $\mu$ = 0
Standard Deviation $\sigma$ = 1

Step 1: Calculate the exponent part:

$-\frac{(x - \mu)^2}{2\sigma^2} = -\frac{(1 - 0)^2}{2 \times 1^2} = -\frac{1}{2} = -0.5$

Step 2: Calculate $e$ raised to the power of the result:

$e^{-0.5} \approx 0.60653$

Step 3: Multiply by the reciprocal of $\sigma \sqrt{2\pi}$ :

$f(x) = \frac{1}{1 \sqrt{2\pi}} \times 0.60653 \approx 0.24197$