Descriptive statistics Experiment  Data  Sample Statistics Experiment  Data  Sample Statistics Sample mean Sample mean Sample variance Sample variance Normalize sample variance by N-1 Normalize sample variance by N-1 Standard deviation goes as square-root of N Standard deviation goes as square-root of N

Inferential Statistics Model Model Estimates of parameters Estimates of parameters Inferences Inferences Predictions Predictions

Importance of the Gaussian

Why is the Gaussian important? Sum if independent observations converge to Gaussian, Central Limit Theorem Sum if independent observations converge to Gaussian, Central Limit Theorem Linear combination is also Gaussian Linear combination is also Gaussian Has maximum entropy for given  Has maximum entropy for given  Least-squares becomes max likelihood Least-squares becomes max likelihood Derived variables have known densities Derived variables have known densities Sample means and variances of independent samples are independent Sample means and variances of independent samples are independent

Derived distributions Sample mean is Gaussian Sample mean is Gaussian Sample variance is distributed Sample variance is distributed Sample mean with unknown variance is Student-t distributed Sample mean with unknown variance is Student-t distributed This allows us to get confidence intervals for mean and variance This allows us to get confidence intervals for mean and variance

Simulating random arrivals Method 1: take small  t, flip coin with event probability   t Method 1: take small  t, flip coin with event probability   t Method 2: generate exponential variable to determine next arrival time (use transformation of uniform) Method 2: generate exponential variable to determine next arrival time (use transformation of uniform)