elec 303 – random signals lecture 18 – statistics, confidence intervals dr. farinaz koushanfar...
TRANSCRIPT
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ELEC 303 – Random Signals
Lecture 18 – Statistics, Confidence IntervalsDr. Farinaz Koushanfar
ECE Dept., Rice UniversityNov 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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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
1)ˆˆ(P nn
]ˆ,ˆ[ nn
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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 X1, X2, X3, ... Xn be a sequence of n independent and identically distributed RVs with finite expectation µ and variance σ2 > 0
• CLT: as the sample size n increases, PDF of the sample average of the RVs approaches N(µ,σ2/n) irrespective of the shape of the original distribution
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CLT
A probability density function Density of a sum of two variables
Density of a sum of three variables Density of a sum of four variables
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CLT
• Let the sum of n random variables be Sn, given by Sn = X1 + ... + Xn. Then, defining a new RV
• The distribution of Zn 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 Xi 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 Xi’s are i.i.d normal, the variance is an estimate and the RV Tn (below) is not normally distributed