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
The logic of confidence intervals

• The mean with unknown variance is distributed as
  Student-t that is, if samples xi are normally
  distributed,
• where is the sample mean and is the
  sample variance, is distributed as Student-t
• Pick q1 and q2 from tables so that
• prob q1 lt lt
  q2 0.99

7
Then
lt µ lt
which gives us confidence intervals on where the
actual mean can be
8
Simulating random arrivals

• Method 1 take small ?t, flip coin with event
  probability ? ?t
• Method 2 generate exponentially distributed r.
  variable to determine next arrival time (use
  transformation of uniform)

9
Binomial distribution (Bernoulli trials)

• Suppose we flip a fair coin n times. The mean
  of heads is n/2, and the standard deviation is
  .
• For large n ( about gt 30), the distribution,
  called
• binomial, approaches normal. Specifically, if
  x is the number of heads, the normalized
  variable
• is
  distributed as N(0,1),
• the normal distribution with mean 0 and
  variance 1.

10

• This enables to estimate probability of events
  using Bernoulli trials very easily.
• Example We flip a coin 100 times and observe 60
  heads. What is the probability of that event?

11
Martin Gardner How not to test a Psychic
(Prometheus, 1989)

• p. 31 report of claim that a psychic subject
  made 781 hits out of 1000. That corresponds to z
  17.8 z
• ----------------
• Notice that we get here is probeventhypothes
  is, where the hypothesis is that the trials are
  Bernoulli.
• What we dont get is the probhypothesisevent.

9.5 ? 1.049 E-21