Descriptive statistics

1
Descriptive statistics

• Experiment ? Data ? Sample Statistics
• Sample mean
• Sample variance
• Normalize sample variance by N-1
• Standard deviation goes as square-root of N

2
Inferential Statistics

• Model
• Estimates of parameters
• Inferences
• Predictions

3
Importance of the Gaussian

4
Why is the Gaussian important?

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

5
Derived distributions

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

6
Simulating random arrivals

• 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)