Lecture Notes on Random Variables and Distributions

Bernoulli Random Variable

• Parameter (p): Probability of success.
• Values: 1 with probability p and 0 with probability 1-p.
• Expectation (E[X]): Equal to p.
• Variance (Var[X]): Calculated as ( p(1-p) ).
  • Example: If p = 10% (such as in a lottery), E[X] = 0.1, Var[X] = 0.1 * 0.9 = 0.09.

Binomial Distribution

• Parameters (n, p): Number of trials (n) and probability of success (p).
• Expectation: ( n \times p ).
• Variance: ( n \times p \times (1-p) ).
  • Example: Playing a game 100 times with a 10% win rate, E[X] = 10, Var[X] = 9.

Geometric Distribution

• Parameter (p): Probability of success on each trial.
• Expectation: ( 1/p ).
• Variance: ( (1-p)/p^2 ).
  • Example: With p = 10%, on average, expect to play 10 times to win once.

Poisson Distribution

• Parameter (λ): Rate or expected number of occurrences.
• Expectation and Variance: Both equal to λ.
  • Used for events with a small probability of occurrence over a large number of trials.

Continuous Uniform Distribution

• Parameters (a, b): Range a to b.
• PDF: ( 1/(b-a) ) for x in [a, b].
• Expectation: ( (a+b)/2 ).
• Variance: ( (b-a)^2/12 ).
  • Example: Waiting time uniformly distributed between 30 and 50 minutes has E[X] = 40.

Exponential Distribution

• Parameter (λ): Rate parameter.
• PDF: ( \lambda e^{-\lambda x} ) for x ≥ 0.
• CDF: ( 1 - e^{-\lambda x} ).
  • Memoryless Property: P(X > s + t | X > s) = P(X > t).
  • Example: With λ = 2, the probability of waiting more than 10 minutes is ( e^{-20} ).

---

Additional Concepts

• Memoryless Property: Particularly relevant for exponential distributions.
• Integration for PDFs: Integral of PDF over its range equals 1.
• Moments and Variance: Calculations often involve integration.

These notes summarize key points from the lecture on random variables, focusing on Bernoulli, binomial, geometric, Poisson, uniform, and exponential distributions, along with their properties and example calculations.