G89.2228%20Lecture%203b

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G89.2228Lecture 3b

• Why are means and variances so useful?
• Recap of random variables and expectations with
  examples
• Further consideration of random variables
• Expected mean and variance of averages
• Estimates of population variance
• Bias of Variance Estimator

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Why are Means and Variances so useful?

• The commonly observed NORMAL distribution is
  indexed by two parameters ? and ?2, the mean
  and variance
• ? is the index of location, and ?2 is the index
  of spread. We can estimate the relative
  frequency of values given m and s2

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An example learning about distributions

• Suppose we were planning to study performance
  variables that are known to be affected by
  anxiety.
• Is the distribution of performance scores
  obtained in the month following the WTC attack
  systematically different from previous studies?
• Suppose we plan to measure performance with a
  measure that goes from 1 to 10, but published
  studies used a measure that ranged from 0 to 5.
  How are the means and variances affected by this
  difference in range?

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Expectations Recap

• A Random Variable is a real-valued function
  defined on a sample space.
• f(X) is a function that describes the likelihood
  of each value of X
• Density function for continuous X
• Probability mass function for discrete X
• Suppose that g(X) is any arbitrary function of
  values of X.
• E(g(X)) is the expectation of g(X), the average
  value of g(X) in the population
• For continuous variables
• For discrete variables

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Recap First Moment(the Mean ?x)

• E(X)?x is the first moment, the mean
• For k an arbitrary fixed constant
• E(Xk) E(X)k ?x k
• E(kX) kE(X) k ?x
• Let Y be a second random variable (perhaps
  related to X, perhaps not)
• E(XY) E(X)E(Y) ?x ?y
• E(X-Y) E(X)-E(Y) ?x - ?y

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Example

• We can relate the 1-10 scale to the 0-5 scale
  with a simple linear function.
• Let X be on the 0-5 scale.
• G(X) is on the 1-10 scale
• G(X) (9/5)X 1
• If E(X), the mean of X, is mX then EG(X), the
  mean of G(X), is
• EG(X) (9/5) mX 1

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Recap Second Moment(the Variance V(X))

• Let k be a fixed constant
• Let Y be another random variable independent of
  X, then

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Example

• If X is on the 0-5 scale.
• G(X) is on the 1-10 scale
• G(X) (9/5)X 1
• If V(X), the variance of X, is s2 X then
  VG(X), the variance of G(X), is
• VG(X) (9/5)2 s2 X
• The standard deviation is the square root of the
  variance
• The standard deviation of G(X) is simply (9/5)
  the standard deviation of X.

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Notes on Random Variables

• Statisticians consider all instances of X to be
  random variables
• E.G., A sample of 10 women measured on CESD gives
  10 random variables
• independent if sampled randomly
• identically distributed if from same population
• hence, same f(X)
• i.i.d. is shorthand for independent, identically
  distributed
• Note that data analysts use the term variable
  to refer to one kind of measure. If the sample
  has n subjects, the variable describes the set of
  n random variables in the statisticians sense.

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Random Variables Need Not Be Independent

• Three outcomes measured on a single subject are
  three random variables
• They are not likely to be independent, nor to
  have the same f(X)
• We would then consider the multivariate joint
  density, f(X1,X2,X3)
• Random variables can be nonindependent in other
  ways
• Unit of analysis issue
• E.g., randomly selected employees within randomly
  selected supervisors teams
• If supervisor level is ignored, employees are not
  sampled randomly (rather in clusters)
• Within a team, the employees may be considered
  independent
• Average team score may be assumed to be
  independent over supervisors, however

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Example Sample of Size 10

• The values at the right have a variance of 3.8,
  (standard deviation of 1.9).
• The mean of the sample is 6.2.
• What can we say about the population from which
  the numbers are sampled?
• What can we say about the sample statistics
  themselves?

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Studying sample statistics using expectation
operators

• Let be the sample average of n random
  variables that are independently sampled from the
  same distribution (i.i.d). (The expected mean of
  each X is the same, as is the expected
  variance).
• Because the expectation of the sample mean is
  equal to the parameter it is estimating, we say
  it is unbiased.

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Expected variance of the sample mean

• The expected variance of the sample mean goes
  down directly with increased sample size, n.

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Bias of a Variance Estimator

• If variance is defined as the average squared
  deviation from the mean, consider the estimate,
• On the average, will this function of the data
  give an unbiased estimate?
• The answer is NO!
• The conceptual reason is that the sample mean is
  itself variable
• The expected value of the above sample estimate
  is ??(n-1)/n

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Bias of a Variance Estimator 2

• First, lets derive an alternative definition of
  variance
• Next, lets do the same for our biased variance
  estimator

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Bias of a Variance Estimator 3

• To determine bias, we determine the expected
  value
• The first term is

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Bias of a Variance Estimator 4

• The second term is
• Hence,
• To make it unbiased,