# Probability Density Function Calculator

**Probability Density Function Calculator** calculates the probability density for the given function. The area under a curve can be determined by performing a definite integral between the given limits.

## What is Probability Density Function Calculator?

Probability Density Function Calculator is an online tool that helps to calculate the probability density for the given function. This online probability density function calculator helps you to calculate the probability density in a few seconds. To use this probability density function calculator, enter the function and limit values in the given input box.

### Probability Density Function Calculator

**NOTE:** Enter function in terms of 'x' only and enter limits upto two digits only.

## How to Use Probability Density Function Calculator?

Please follow the steps below to find the probability density using an online probability density function calculator:

**Step 1:**Go to Cuemath’s online probability density function calculator.**Step 2:**Enter the function, and limits values in the given input box of the probability density function calculator.**Step 3:**Click on the**"Calculate"**button to find the probability density for the given function.**Step 4:**Click on the**"Reset"**button to clear the fields and enter the new values.

## How Probability Density Function Calculator Works?

The** probability density function** is defined as the probability function represented for the density of a continuous random variable that falls within a specific range of values. The formula to calculate the probability density function is given by

PDF = \(\int\limits_a^b {f\left( x \right)dx}\)

The fundamental theorem of calculus tells us that to calculate the area under a curve y = f(x) from x = a to x = b. It is represented as

We first calculate the integration g(x) of f(x), \(g\left( x \right)= \int {f\left( x \right)dx}\) and then evaluate g(b) − g(a). That is, the area under the curve f(x) from x = a to x = b is \(\int\limits_a^b {f\left( x \right)dx = g\left( b \right) - g\left( a \right)}\)

Let us understand this with the help of the following example.

**Solved Examples on Probability Density Function Calculator**

**Example 1:**

Find the probability density function for f(x) = x^{2} + 2x for the interval [1, 3] and verify it using the probability density function calculator

**Solution:**

Given: f(x) = x^{2} + 2x

\(Area = \int_{a}^{b}f(x)\)

\(=\int_{1}^{3}(x^2 + 2x)\)

= 16.67

**Example 2:**

Find the probability density function for f(x) = 3x^{2} - 4x for the interval [2, 4] and verify it using the probability density function calculator

**Solution:**

Given: f(x) = 3x^{2} - 4x

\(Area = \int_{a}^{b}f(x)\)

\(=\int_{2}^{4}(3x^2 - 4x)\)

= 32

Similarly, you can try the probability density function calculator and find the area for:

- f(x) = 5x + 6 for limits x = -3 to 1
- f(x) = x
^{3}/ 2 for limits x = 2 to x = 5

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