Lecture Notes: Variance of a Random Variable

Overview

• Topic: Variance of a random variable
• Builds on previous concepts of sample variance and theoretical mean.
• Focus on theoretical variance using a probability distribution.

Key Concepts

Different Formulas for Variance

• Technical Formulas: Various formulas available for calculating variance.
• Theorem in Textbook: Offers a preferred equation.
• Personal Preference: Suggested one formula as easier, but encourages trying others.

Variance Representation

• Graphical Representation:
  • Smaller variance shows mean with closer data points.
  • Larger variance shows mean with data points spread out.

Discrete vs. Continuous Data

• Discrete Data: Probabilities in buckets, histogram style.
• Continuous Data: Smooth curve in probability distribution.

Example Problems

Example 1: Discrete Data (Defective Parts)

• Random Variable X: Number of defective parts in a sample.
• Probability Distribution Provided: Values for X and F(X).
• Formulas Used:
  • (\sigma^2 = \sum (X - \mu)^2 f(X))
  • Calculate mean (\mu) as expected value of X.
  • (\mu = 0.61)
  • Variance (\sigma^2 = 0.4979)
  • Standard Deviation (\sigma = 0.7056)

Example 2: Continuous Data (Water Demand)

• Random Variable X: Weekly demand for drinking water.
• Probability Density Function: (f(x) = 2(x-1)) from 1 to 2.
• Integral Method Used:
  • Mean calculation using integral.
  • (\mu = \frac{5}{3})
  • (\sigma^2 = \frac{1}{18})

Example 3: Discrete Data (Function of X)

• Function g(x): (g(x) = 2x + 3)
• Probability Distribution Provided
• Formulas Used:
  • Mean (\mu_{g(x)} = 6)
  • Variance of (g(x) = 4)_

Important Reminders

• Always Square Mu in Variance Calculation: Common mistake to miss squaring.
• Integrals Practice: Encouraged to practice integrals to improve problem-solving speed.
• Choose Formula Based on Data Given: Different formulas might be more convenient depending on the data type.

Conclusion

• Both discrete and continuous data have distinct methods for calculating variance.
• Practice and familiarity with different formulas and integrals is crucial for efficiency.