elec 303 – random signals lecture 18 – statistics, confidence intervals dr. farinaz koushanfar...

ELEC 303 – Random Signals

Lecture 18 – Statistics, Confidence IntervalsDr. Farinaz Koushanfar

ECE Dept., Rice UniversityNov 10, 2009

Statistics

Example

Reduction of Cholesterol Level

Example (Cont’d)

Sample Mean

Sample Median

Sample Median (Cont’d)

Sample Mean vs. Sample Median

Percentile

Location of Data

Variability

Averages

Sample Variance

Statistics

Standard Deviation

Sample Range

Interquartile Range

Averaging?

Data Handling

Dot Plots

Histogram

Example

Histogram (Cont’d)

Histogram (Cont’d)

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

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

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

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

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

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

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:

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