Sta 113, fall 2006

Probability and Statistics for Probability and Statistics for Engineering Engineering

Instructor: Sayan MukherjeeTAs: N. Pillai, H. Wang

STAT 113

Sta 113, fall 2006

There are three kinds of lies: lies, damned lies, and statistics.

B. Disraeli

Perspectives on stats

Sta 113, fall 2006

What is probability ?What is probability ?

Probability is a branch of mathematics that deals with calculating the likelihood of a given event's occurrence, which is expressed as a number between 1 and 0.

Sta 113, fall 2006

What is statistics ?What is statistics ?

Statistics derives from: Latin -- statisticum collegium ("council of state")Italian -- statista ("statesman" or "politician").

Statistik: German first introduced by Gottfried Achenwall (1749), originally designated the analysis of data about the state, or the "science of state". Acquired the meaning of the collection and classification of data generally in the early 19th century.

Statistics as inverse probability -- estimating parameters from experimental data

Sta 113, fall 2006

Well-posed problems

A problem is well-posed if its solution

• exists

• is unique

• is stable, eg depends continuously on the data

Inverse problems are typically ill-posed

Sta 113, fall 2006

Class requirements and rulesClass requirements and rules

Course webpage

Sta 113, fall 2006

First digitsFirst digits

http://en.wikipedia.org/wiki/List_of_world_records

Count entries starting with: {1,2,3,4,5,6,7,8,9}

Count entries ending with: {1,2,3,4,5,6,7,8,9}

Accounting fraud

Sta 113, fall 2006

What’s wrong with the heartland ?What’s wrong with the heartland ?

Sta 113, fall 2006

It’s the emptinessIt’s the emptiness

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The geometry of randomness

Dido’s problem (Isoperimetry) : Among all closed level curves of fixed length, find the one that encloses the largest area.

A

A

Sta 113, fall 2006

The geometry of Gaussian random variables

A Gaussian distribution:

Sta 113, fall 2006

The geometry of Gaussian random variables

A draw of n Gaussian random variables is a point in an n-dimensional space. How far from the origin is this point ?

x x12 x2

2 ... xn2

For n large the answer is that with very high probability

1c

nx

n1

c

n

Sta 113, fall 2006

Law of large numbers or central limit theorem

The previous observation is a special case of the following phenomena:

Given a smooth function of n variables

x (x1,...,xn ) the following is true

Pr f x x f x h C1 exp C2h2n .

A classic example : f (x) x1 x2 ... xnn

.

Sta 113, fall 2006

Regression -- pedestrian detection

Papageorgiou and Poggio, 1998

Sta 113, fall 2006

Daimler ChryslerDaimler Chrysler

Sta 113, fall 2006

Experimental MercedesExperimental Mercedes

A fast version, integrated with a real-time obstacle

detection system

MPEG

Constantine Papageorgiou

Sta 113, fall 2006

People classification/detectionPeople classification/detection

Stuttgart

STA 293 03, fall 2005

More regression: talking faces

Text-to-visual-speech (TTVS) systems:

Movies in faces directory

STA 293 03, fall 2005

More regression: talking faces

Text-to-visual-speech (TTVS) systems:

Movies in faces directory

STA 293 03, fall 2005

Descriptive statistics and visualization

Click on mandarin in visual

Click on mandarin in visual

STA 293 03, fall 2005

Conclusion

Statistics is about predictive modeling that quantifies uncertainty

There are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know.

---- Donald Rumsfeld