Marginal Probability Density Functions (PDFs)

Overview

• Focus: Marginal probability density functions for discrete random variables.
• Goal: Understand the concept using examples.

Definitions

• Discrete Random Variables: Variables like x and y with specified range spaces (R_x and R_y).
• Joint Probability Density Function: Denoted as f(x, y).
• Marginal Probability Density Function of x: Denoted as f_1(x), calculated by summing over all values of y.
  • Formula: ( f_1(x) = \sum_{y} f(x, y) )
• Marginal Probability Density Function of y: Calculated by summing over all values of x.
  • Formula: ( f_2(y) = \sum_{x} f(x, y) )

Example Context

• A group of 9 coworkers: 4 with PhD, 3 with Master's, 2 with Undergraduate.
• Scenario: Randomly promote 3 people.
  • x = Number of PhDs promoted.
  • y = Number of Master's promoted.
• Joint PDF: Created based on the above scenario.

Calculation Steps

Calculating Marginal PDF of x

1. Range of y: Values from 0 to 3 (Number of Master's promoted).
2. Formula: ( f_1(x) = f(x, 0) + f(x, 1) + f(x, 2) + f(x, 3) )
3. Example: For x = 1 (1 PhD promoted):
   • Calculate f(1, 0), f(1, 1), f(1, 2), f(1, 3)
   • Values: 4/84, 24/84, 12/84, 0 respectively.
   • Total Probability: ( 40/84 ) for 1 PhD being promoted.

Calculating Marginal PDF of y

1. Range of x: Values from 0 to 3 (Number of PhDs promoted).
2. Formula: ( f_2(y) = f(0, y) + f(1, y) + f(2, y) + f(3, y) )
3. Example: For y = 2 (2 Master's promoted):
   • Calculate f(0, 2), f(1, 2), f(2, 2), f(3, 2)
   • Values: 6/84, 12/84, 0, 0 respectively.
   • Total Probability: ( 18/84 ) for 2 Master's being promoted.

Interpretation

• Marginal of x: Probability of a certain number of PhDs being promoted, ignoring Master's.
• Marginal of y: Probability of a certain number of Master's being promoted, ignoring PhDs.
• Conceptual Understanding: Marginalization helps focus on one variable by summing over the other.

Additional Example

• Probability of 1 Master's promoted:
  • Calculated by summing: 3/84 + 24/84 + 18/84 + 0 = ( 45/84 )

Conclusion

• Purpose of Marginals: Simplify joint distributions to focus on individual random variables.
• Application: Useful in statistical analysis where simplification of complex relationships is needed.