Introduction to the Normal Distribution

Overview

• Normal Distribution (also known as Gaussian distribution) is a critical continuous probability distribution.
• Arises frequently in both theoretical contexts and real-world scenarios.

Key Characteristics

• Probability Density Function (PDF): Describes the height of the curve at any point x, denoted as f(x).
• Equation: Important but not frequently used directly in basic applications.

Parameters of the Normal Distribution

• Random Variable X: Can take any finite value.
• Mu (µ): Mean of the distribution, can also take any finite value.
• Sigma (σ): Standard deviation, must be a positive value.
  • Variance (σ²): Square of the standard deviation, also positive.

Properties of the Distribution

• Symmetry: The distribution is symmetric about the mean (µ).
  • Mean (µ) is both the median and the center of the distribution.

Standard Deviation Significance

• One Standard Deviation: ~68% of data falls within one standard deviation of the mean.
• Two Standard Deviations: ~95% of data falls within two standard deviations.
• Three Standard Deviations: ~99.7% of data falls within three standard deviations.

Visual Representation

• Comparison of Standard Deviations:
  • Red line: Normal distribution with σ_2 (double the σ_1).
  • White curve: Normal distribution with σ_1.
  • Greater σ results in more area under the tails and a lower peak.

Notation

• If X is normally distributed with mean µ and variance σ², it's denoted as:
  • X ~ N(µ, σ²)
• Be cautious: different notations may use standard deviation instead of variance for the second term.

Standard Normal Distribution

• Definition: Normal distribution with mean of 0 and variance of 1.
• Notation: Z ~ N(0, 1), where Z is a random variable with standard normal distribution.
• Variance of 1 implies σ = 1.

Practical Applications

• Probability Calculations:
  • Probabilities are areas under the normal distribution curve.
  • Requires integrating the density function; no closed-form solution.
  • Use of software or standard normal tables recommended for practical computations.