Lecture Notes: Continuous Probability Distributions and Normal Distribution

Introduction to Chapter Six

• Focus on continuous probability distributions
• Skip initial section to dive into the normal distribution

Understanding Normal Distribution

• Models distribution of processes as the sum of component processes
• Describes phenomena in nature, industry, and research
  • Examples: Meteorological experiments, rainfall studies, measurements of manufactured parts
• Basis for topics like confidence intervals and hypothesis testing

Characteristics of Normal Distribution

• Continuous Variables
  • Can take on decimal forms, not just whole numbers
  • Requires integration, not summation
• Z Table
  • Used for standard normal curve calculations
  • Z values relate to standard normal curve

Properties of a Normal Curve

• Mode (µ): Point where curve is a maximum; it is also the median
• Symmetry: Curve is symmetric about the vertical axis through the mean
• Points of Inflection: Occur at x = µ ± σ
  • Concave downward between µ - σ and µ + σ
  • Concave upward otherwise
• Asymptotic Behavior
  • Approaches horizontal axis asymptotically
• Total Area Under Curve
  • Equals 1, representing total probability

Comparing Normal Curves

• Curves can have different mean (µ) and standard deviation (σ)
• Overlapping curves: Same σ, different µ
• Varying steepness indicates differences in σ

Calculating Probabilities with Normal Distribution

• Continuous distributions: Probability of exact value is 0
• Use of Z values to standardize µ and σ
  • Converts x to z using: z = (x - µ) / σ
  • Standard normal curve: µ = 0, σ = 1
• Z Table: Provides standard normal probabilities
  • Z values represent number of standard deviations from the mean

Example Problems

Problem 1: Probability Right of Z value

• Find area under curve to the right of Z = 1.84
  • Use Z table to find probability

Problem 2: Probability Between Two Z values

• Find area between Z = -1.97 and Z = 0.86
  • Subtract lower Z value probability from higher Z value probability

Problem 3: Non-Standard Normal Distribution

• Given µ = 50, σ = 10
• Find probability that X is between 45 and 62
  • Convert X to Z using formula
  • Calculate probabilities using Z table

Conclusion

• Normal distribution is foundational, more practice and topics to follow