Enter the point, mean, and standard deviation into the calculator to determine the probability density function.

Probability Density Formula

The following formula is used to calculate the probability density function for a continuous random variable.

f(x) = 1 / (σ * √(2π)) * e^{(-(x - μ)^2 / (2σ^2))}


  • f(x) is the probability density function x is the point for which you want to find the probability μ is the mean or expectation of the distribution (also the median and mode) σ is the standard deviation e is the base of the natural logarithm, approximately equal to 2.71828

To calculate the probability density, subtract the mean from the point for which you want to find the probability and square the result. Multiply this by -1 and divide by twice the square of the standard deviation. Raise e (the base of the natural logarithm) to the power of this result. Multiply this by the reciprocal of the product of the standard deviation and the square root of twice pi.

What is a Probability Density?

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.

How to Calculate Probability Density?

The following steps outline how to calculate the Probability Density using the given formula:

  1. First, determine the value of x, the point for which you want to find the probability.
  2. Next, determine the values of μ (mean) and σ (standard deviation) for the distribution.
  3. Next, substitute the values of x, μ, and σ into the formula: f(x) = 1 / (σ * √(2π)) * e^(-(x – μ)^2 / (2σ^2)).
  4. Finally, calculate the probability density by evaluating the formula.
  5. After inserting the variables and calculating the result, check your answer with the calculator or statistical software.

Example Problem:

Use the following variables as an example problem to test your knowledge:

x = 2

μ = 5

σ = 1.5