730 Lecture 2

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730 Lecture 2

• Todays lecture
• Inequalities
• Convergence
• Taylor Series

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Inequalities
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Chebychev
Area
Area s2/e2
m-e m me
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Chebychev proof
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Jensen
g(x)
m
m
g(m) g(m)(x-m)
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Example sample standard deviation

• Graph of yÖx

s2
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Std dev (cont)

• Slope at xs2 is 1/2s
• Tangent at point (s,s2) lies above curve
• THUS

Sample sd BIASSED DOWN!
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Convergence

• Convergence in distribution
• Main use approximation Eg CLT

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Examples

• tn converges to normal
• Binomial converges to Poisson
• Gamma converges to normal

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Central limit Theorem (CLT)
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Example

• Convergence of binomial to Normal
• Express Xn as

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Example (cont)

• Then result is just CLT, as the Ys have
• mean p and variance p(1-p).

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Example

• Convergence of binomial to Poisson
• Proved by showing convergence of the prob.
  function since

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Example 5 (cont)
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Convergence (2)

• Convergence in probability Xn
• converges to X in probability if

• Eg proportion of heads in n
• tosses 0.5

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Example

• Xnproportion of heads in n tosses
• E(Xn)p0.5, Var(Xn)p(1-p)/n1/4n
• By Chebychev,
• P(Xn-0.5gte) Var(Xn)/e2 1/(4n e2) 0

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More examples

• Sample mean converges in probability to
  population mean as sample size gets larger
• (Weak law of large numbers)
• Sample variance converges to population variance
• Consistent estimators

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Example sample variance
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Sample variance (cont)
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Example

• X1,Xn Uniform
• Mnmax (X1,Xn )
• P(Mn x) xn (why?)
• P(Mn- 1gte) P(Mnlt1- e) (1- e)n 0
• Thus Mn converges to 1 in probability