Probability Density Function with Normal Distribution: Explained

Understanding the Probability Density Function with Normal Distribution

The normal distribution, also known as the Gaussian distribution, is one of the most important probability distributions in statistics. It is often used because many natural phenomena tend to follow this distribution pattern. The Probability Density Function (PDF) of a normal distribution provides insight into the likelihood of different outcomes within the dataset. The PDF formula for a normal distribution is broken down below and explained step by step for clarity.

Formula Breakdown

• x is the variable whose probability density you want to find (measured in the same units as the mean and standard deviation, e.g., income in USD, height in meters).
• mu (μ) is the mean or average of the distribution (measured in the same units as the variable x).
• sigma (σ) is the standard deviation of the distribution (measured in the same units as the variable x).

Example Description

Imagine you are an analyst at a company studying employee salaries, which follow a normal distribution. You have a mean salary (mu) of $50,000 and a standard deviation (sigma) of $10,000. You want to find out the probability density of an employee earning exactly $60,000. Plugging these values into our formula:

f(60000, 50000, 10000) = (1 / (10000 * Math.sqrt(2 * Math.PI))) * Math.exp( 0.5 * Math.pow((60000   50000) / 10000, 2))

This gives us the probability density at $60,000.

Output

• The result f(x, mu, sigma) is expressed in terms of probability density.

Example Valid Values

• For x = 60000, mu = 50000, sigma = 10000
• For x = 55, mu = 50, sigma = 5

How PDF Helps in Real Life Scenarios

The Probability Density Function helps us understand the behavior of different variables, allowing for better decision making in a variety of fields. For example, in finance, it helps in risk management by evaluating the likelihood of stock prices hitting certain levels. In biology, it assists in understanding traits within a population, such as height or blood pressure levels.

Data Validation

All inputs must be valid real numbers for meaningful results. The standard deviation, sigma, must be greater than zero.

Summary

The Probability Density Function for a normal distribution is a powerful statistical tool that provides insight into the probability of various outcomes in a dataset. This article explains the formula, its components, and provides detailed descriptions, example values, and practical applications to make the concept understandable and applicable.