CIS 2033 A Modern Introduction to Probability and Statistics Understanding Why and How

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CIS 2033A Modern Introduction to Probability and
StatisticsUnderstanding Why and How

• Chapter 17 Basic Statistical Models
• Slides by Dan Varano
• Modified by Longin Jan Latecki

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17.1 Random Samples and Statistical Models

• Random Sample A random sample is a collection of
  random variables X1, X2,, Xn, that that have the
  same probability distribution and are mutually
  independent
• If F is a distribution function of each random
  variable Xi in a random sample, we speak of a
  random sample from F. Similarly we speak of a
  random sample from a density f, a random sample
  from an N(µ, s2) distribution, etc.

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17.1 continued

• Statistical Model for repeated measurements
• A dataset consisting of values x1, x2,, xn of
  repeated measurements of the same quantity is
  modeled as the realization of a random sample X1,
  X2,, Xn. The model may include a partial
  specification of the probability distribution of
  each Xi.

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17.2 Distribution features and sample statistics

• Empirical Distribution Function
• Fn(a)
• Law of Large Numbers
• lim n-gt8 P(Fn(a) F(a) gt e) 0
• This implies that for most realizations
• Fn(a) F(a)

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17.2 cont.

• The histogram and kernel density estimate
• f(x)
• Height of histogram on (x-h, xh f(x)
• fn,h(x) f(x)

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17.2 cont.

• The sample mean, sample median, and empirical
  quantiles
• ?n µ
• Med(x1, x2,, xn) q0.5 Finv(0.5)
• qn(p) Finv(p) qp

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17.2 cont.

• The sample variance and standard deviation, and
  the MAD
• Sn2 s2 and Sn s
• MAD(X1, X2,,Xn) Finv(0.75) Finv (0.5)

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17.2 cont.

• Relative Frequencies for a random sample X1,X2,
  . . . , Xn from a discrete distribution with
  probability mass function p,one has that
• p(a)

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17.4 The linear regression model

• Simple Linear Regression Model In a simple
  linear regression model for a bivariate dataset
  (x1, y1), (x2, y2),,(xn, yn), we assume that x1,
  x2,, xn are nonrandom and that y1, y2,, yn are
  realizations of random variables Y1, Y2,, Yn
  satisfying
• Yi a ßxi Ui for i 1, 2,, n,
• Where U1,, Un are independent random variables
  with EUi 0 and Var(Ui) s2

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17.4 cont

• Y1, Y2,,Yn do not form a random sample. The Yi
  have different distributions because every Yi has
  a different expectation
• EYi Ea ßxi Ui a ßxi EUi a
  ßxi