Review

• Exam 1 being returned.
• Red Sox probability problem: what's the probability that they will win, given that they are 2-0 in the series and assuming that the teams are equally matched.
• Electoral College maps versus regular area maps. For example:

  

• Global Heat and Cold Rubidium -- questions?
• Discrete Event Simulation
  1. Let's try the M/M/1 queue solution

     M/M/1

  2. Let's estimate the queue length for a given pair of parameters
  3. Let's try a Grocery Store:

     grocery

  4. Eventually, we'd like to establish a confidence interval for that estimate. But we have some work to do, first.

Sampling Distributions

• Key point: the sampling distribution is not the same as the population. It is the distribution of values computed on samples from the distribution.
• We'll do an exercise with this.

Standard Error of the Mean

Very often the statistic we are computing on samples in the mean of the sample. The following two facts hold:

• Mean of sampling distribution of the mean (x-bar) equals the population mean (μ)
• The variance of the sampling distribution of the mean is less than the variance of the population:

  σ_x-bar = σ/sqrt(n)

• Special name: standard error

The Central Limit Theorem

Amazingly enough:

The distributions of sums (and therefore means) is increasingly Gaussian

We'll explore the following model

CLT2

Limitations of the CLT:

• The samples must be independent
• The factors must be additive in effect. That is, if some measurement is the result of many independent factors, we might want to invoke the CLT and assume it is Gaussian, but not if the factors have a synergistic effect, say. Example: the area of a rectangle is not an additive result of the width and height.
• The CLT says that the distribution approaches the Gaussian, but it doesn't say how fast. In practice, it's amazingly fast, but if you really need high accuracy, you wouldn't want to rely on the CLT.