Random Variables Lecture Notes

Introduction to Random Variables

• Random Experiment: An experiment where the outcome is uncertain.
• Sample Space (S): The set of all possible outcomes of a random experiment.
• Event: A subset of the sample space.
• Probability: Defined using the axiomatic approach.

Overview of Topics Covered

• Definition of random variables.
• Types of random variables: Discrete and continuous.
• Probability Mass Function (PMF): Distribution for discrete random variables.
• Cumulative Distribution Function (CDF): Distribution for random variables.
• Concepts of Expectation and Variance of random variables.

Definition of a Random Variable

• A random variable is a numerical quantity associated with the outcomes of a random experiment.
• Example: Rolling a die twice can produce outcomes, but if only the sum of outcomes is of interest, that sum is defined as a random variable.

Example: Rolling a Die Twice

• Rolling outcomes: (1,1), (1,2), ..., (6,6).
• Numerical quantities of interest:
  • Sum (X): Defined as the total from both rolls.
  • Smaller Outcome (Y): The smaller number from the two rolls.
• Possible values for X: 2 through 12.
• Possible values for Y: 1 through 6.

Assigning Probabilities to Random Variables

• The value of a random variable can be assigned probabilities based on the relevant events from the sample space.

• Example of X (Sum of Rolls):

  • P(X=2) = 1/36
  • P(X=3) = 2/36
  • P(X=4) = 3/36
  • Continuing this pattern...
  • P(X=12) = 1/36

• Example of Y (Smaller Outcome):

  • P(Y=1) = 11/36
  • P(Y=2) = 9/36
  • P(Y=3) = 7/36
  • P(Y=4) = 5/36
  • P(Y=5) = 3/36
  • P(Y=6) = 1/36

Additional Example: Tossing a Coin Three Times

• Possible outcomes for three tosses: 8 total outcomes (H, H, H), (H, H, T), ..., (T, T, T).
• Define random variables:
  • X: Number of Heads (0 to 3).
  • Y: Toss in which a head appears first (1, 2, 3, or nil).

Assigning Values and Probabilities

• For X:
  • P(X=0) = 1/8
  • P(X=1) = 3/8
  • P(X=2) = 3/8
  • P(X=3) = 1/8
• For Y:
  • P(Y=1) = 4/8
  • P(Y=2) = 2/8
  • P(Y=3) = 1/8
  • P(Y=nil) = 1/8

Conclusion

• Random variables are essential in understanding outcomes of random experiments.
• They help in computing probabilities for various events associated with the outcomes.
• Understanding PMF and CDF is crucial for analyzing discrete and continuous random variables.