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ELEC 303 – Random Signals Lecture 18 – Statistics, Confidence Intervals Dr. Farinaz Koushanfar ECE Dept., Rice University Nov 10, 2009

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Statistics

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Example

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Reduction of Cholesterol Level

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Example (Cont’d)

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Sample Mean

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Sample Median

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Sample Median (Cont’d)

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Sample Mean vs. Sample Median

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Percentile

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Location of Data

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Variability

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Averages

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Sample Variance

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Statistics

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Standard Deviation

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Sample Range

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Interquartile Range

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Averaging?

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Data Handling

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Dot Plots

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Histogram

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Example

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Histogram (Cont’d)

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Confidence interval Consider an estimator for unknown We fix a confidence level, 1- For every replace the single point estimator with a lower estimate and upper one s.t. We call, a 1- confidence interval

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Confidence interval - example Observations Xi’s are i.i.d normal with unknown mean and known variance /n Let =0.05 Find the 95% confidence interval

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Confidence interval (CI) Wrong: the true parameter lies in the CI with 95% probability…. Correct: Suppose that is fixed We construct the CI many times, using the same statistical procedure Obtain a collection of n observations and construct the corresponding CI for each About 95% of these CIs will include

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A note on Central Limit Theorem (CLT) Let X 1, X 2, X 3,... X n be a sequence of n independent and identically distributed RVs with finite expectation µ and variance σ 2 > 0 CLT: as the sample size n increases, PDF of the sample average of the RVs approaches N(µ,σ 2 /n) irrespective of the shape of the original distribution

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CLT A probability density functionDensity of a sum of two variables Density of a sum of three variablesDensity of a sum of four variables

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CLT Let the sum of n random variables be S n, given by S n = X 1 +... + X n. Then, defining a new RV The distribution of Z n converges towards the N(0,1) as n approaches (this is convergence in distribution),written as In terms of the CDFs

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Confidence interval approximation Suppose that the observations X i are i.i.d with mean and variance that are unknown Estimate the mean and (unbiased) variance We may estimate the variance /n of the sample mean by the above estimate For any given , we may use the CLT to approximate the confidence interval in this case From the normal table:

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Confidence interval approximation Two different approximations in effect: – Treating the sum as if it is a normal RV – The true variance is replaces by the estimated variance from the sample Even in the special case where the X i ’s are i.i.d normal, the variance is an estimate and the RV T n (below) is not normally distributed

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