Random Variables and Common Distributions

1. From Outcomes to Variables

A random variable maps experiment outcomes to numerical values.

  • Discrete variables use PMF
  • Continuous variables use PDF/CDF

2. CDF Properties

F(x)=P(X<=x) is: - monotone non-decreasing - right-continuous - tends to 0 at -infinity and 1 at +infinity

3. Important Distributions

Bernoulli / Binomial

Useful for success/failure modeling and repeated trials.

Poisson

Models event counts in fixed interval with rate lambda.

Normal

Central model for aggregated noise due to CLT.

Exponential

Waiting time model in Poisson process.

4. Moments

  • mean
  • variance
  • skewness/kurtosis (shape diagnostics)

5. Worked Examples

  1. If X~Binomial(10,0.4), compute E[X]=4, Var(X)=2.4.
  2. If X~Poisson(5), P(X=0)=e^-5.

6. CS Use Cases

  • failures per hour (Poisson)
  • latency jitter (normal approximation)
  • retries to success (geometric)

Exercises

  1. Compute CDF from PMF for a 3-point discrete variable.
  2. Derive variance of Bernoulli.
  3. Simulate Poisson arrivals and compare empirical histogram.
  4. Explain when normal approximation to binomial is poor.