Lecture Notes on Random Variables and Probability Distributions

Definition of Random Variables

• Random variables can be defined in various ways.
• Example: Coin flip random variable
  • Map the outcomes of a coin flip to numbers.
  • Example: 0 for tails, 1 for heads.

Missing Probability Information

• Definition alone doesn’t provide probability of outcomes (0 or 1).
• Need a Probability Distribution to obtain this information.

Discrete vs. Continuous Random Variables

• The coin flip example is a discrete random variable.
  • Outcomes are distinct and countable.

Discrete Probability Distribution

• Visual representation is common.
• Horizontal axis: Possible outcomes (0 or 1).
• Vertical axis: Probability of each outcome.

Example Probability Distribution

• Outcome 0 probability: Initially stated as 0.5 (50%).
• Outcome 1 probability: Initially stated as 0.6 (60%).
• Issue: Probabilities sum to 1.1, which is not legitimate (should sum to 1).

Correction of Probability Distribution

• Corrected probabilities:
  • Outcome 0 probability: 0.4 (40%).
  • Outcome 1 probability: 0.6 (60%).
• Legitimate probability distribution now as it sums to 1.

Interpretation of Coins and Outcomes

• The coin is not fair due to unequal probabilities.
• More likely to get heads (random variable = 1).

Next Steps

• Further study of various probability distributions.
• Recognizing the importance in inferential statistics.