Random Variables and Distributions

1. Random Variables

A random variable (RV) maps outcomes to numeric values.

  • Discrete RV: countable support, PMF p(x)=P(X=x)
  • Continuous RV: density f(x), with probabilities as integrals

CDF always defined: F_X(t)=P(X<=t).

2. Core Distribution Families

Discrete

  • Bernoulli(p)
  • Binomial(n,p)
  • Geometric(p)
  • Poisson(lambda)

Continuous

  • Uniform(a,b)
  • Normal(mu,sigma^2)
  • Exponential(lambda)

3. Theorem (Poisson Approximation)

If n large and p small with np=lambda, then Binomial(n,p) approximates Poisson(lambda).

Proof sketch from limit of binomial PMF.

4. Memoryless Property

Only geometric (discrete) and exponential (continuous) are memoryless:

P(X>s+t|X>s)=P(X>t).

5. Worked CS Examples

  1. Retries until success -> geometric model.
  2. Request arrivals per minute -> Poisson model.
  3. Sensor noise -> often normal approximation.

6. Distribution Selection Checklist

  • Is variable count-like or real-valued?
  • Is support bounded or unbounded?
  • Is process arrival-based?
  • Is approximation acceptable?

Exercises

  1. Compute mean and variance of Binomial(20,0.3).
  2. Evaluate P(X>=3) for Poisson(2).
  3. Prove CDF of any RV is non-decreasing and right-continuous.
  4. Show exponential is memoryless by direct calculation.