Download Probability&statistics

Probability Theory
Instructor:
Assoc. Prof. Dr. Deshi Ye ( 叶德仕 )
College of Computer Science
Zhejiang University
Email: [email protected]
Course homepage: Http://www.cs.zju.edu.cn/people/yedeshi/prob12/
Outline
 Brief introduction to the course
 Syllabus, course policies and contents
 Introduction to probability and statistics
 History and importance
 Treatment of data
 Graphs: Pareto Diagram, Dot Diagram, Box-plot
 Frequency distribution, Stem-and-leaf Displays
Course information
 What is for?
 This course provides an elementary
introduction to probability with applications.
 Topics include:
 axioms of probability;
 basic probability concepts and models (counting
methods , conditional probability, Bayes
theorem,et.);
 random variables;independence;
 discrete and continuous probability distributions;
 calculate mathematical expectation and variance;
 limit theory
Course Goals
 Students at the end of course should
be able to do the following:
 1) Understand the concepts and methods
of probability theory
 2) Contrast, evaluate, and implement
simulations or experiments
 3) Utilize Minitab program for analyzing
data and summarizing
Syllabus
 Prerequisite: one year course in calculus
 Textbooks (required):
Miller & Freund's Probability and Statistics for
Engineers (Seventh Edition), Richard A. Johnson.
Publishing House of Electronics Industry or Pearson
Education Press.
 Chapter 1-6 for “Probability Theory”, Chapter 7-13 for
the second semester (“Mathematical Statistics”).
 References:
1) A First course in Probability (6th Ed), Sheldon Ross.
China Statistics Press.
2) Probability & Statistics for Engineers & Scientists
(7th Ed), R.E. Walpole,R.H. Myers, S.L. Myers, K. Ye.
Tshinghua or Pearson Education Press.
Grading
 Grades for the course will be based
on the following weighting
1) Class attendance and Homework
assignment: 36%
2) Unit quiz: 24% (12%, 12%)
3) Final exam: 40%
Homework
 1) You may collaborate on homework, but
you must write your submitted work in your
own words. All steps are required, this
includes showing calculations, derivations,
and proofs. Solutions will be posted on the
class web site.
 2)Assignments are due in class as noted in
the syllabus and web page.
Checking web page
 I am highly recommend that each
student check this web page at least
once a week for new announcements
and homework assignments.
http://www.cs.zju.edu.cn/people/yedeshi/software/MiniTAB14.iso
Probability in CS
 Randomized algorithms
 Querying Theory
 Software testing
 Computer simulation and modeling
Introduction
 Probability theory is devoted to the
study of uncertainty and variability
 Probability quantifies how uncertain
we are about future events
 Statistics can be described as the
study of how to make inference and
decisions in the face of uncertainty
and variability
Uncertainty Events






Say red
Coin toss
Matching games (Cards, Name)
Traffic light
The life of a light
Lotteries?
Poker Lotteries
 http://www.zjlottery.com/news/showmes.a
sp?newsid=9950
 Heart, Spade, Club, Diamonds
 1（A）、2、3、4、5、6、7、8、9、10、11
（J）、12（Q）、13（K）
 Arbitrarily choose one piece
 cost 2￥，if win you are awarded 13￥ (win
in 1/13)
Why measure uncertainty?
 To make tradeoffs among uncertain
events
 Measure combined effect of several
uncertain events
 To communicate about uncertainty
Brief History
 Blaise Pascal and Pierre de Fermat:
the origins of probability are found.
 concerning a popular dice game
 fundamental principles of probability
theory
 Pierre de Laplace:
 Before him, concern on the analysis of
games of chance
 Laplace applied probabilistic ideas to
many scientific and practical problems
History cont.
 Mathematical statistics is one important
branch of applied probability; other
applications occur in such widely different
fields as genetics, psychology, economics,
engineering, computer science.
 Important workers: Chebyshev, Markov,
von Mises, and Kolmogorov
 One of the difficulties is the definition of
probability. 20th century, it was solved by
treating probability theory on an axiomatic
basis (Kolmogorov).
Words for probability
 Chance: the falling out or happening
of events
 Stochastic: randomly determined
 Random: not sent or guided in a
special direction, having no definite
aim or purpose
 Aleatory: dependent on the throw of
a die
 Hazard: a chance or venture.
Importance of Prob. Theory
 Two major applications of Prob.
 Risk assessment (new medical treatments)
 Reliability (weather prediction, earthquake, reduce
failure of consumer product)
 Why statistics and probability in engineering?
 Quantify the uncertainty associated with engineer
model
 Evaluate the result of experiment
 Assess importance of measurement uncertainty
 Safeguard for persons, qualities of environment,
assets
A case study
 Visually inspecting data to improve product
quality
 Monitoring manufacturing data
 Ceramic part in coffee makers,
which is made by filling the
mixture of clay-water-oil.
The depth of the slot is uncontrolled.
 Slot depth was measured on three ceramic parts
selected from production every half hour during
the first 6 AM to 3 PM.
Time series plot
Stable: 217.5
Good quality:
[215, 220]
Ch2: Treatment of data
 Outline





Pareto diagrams, dot diagrams
Histograms (Frequency distributions)
Stem-and-leaf display
Box-plot (Quartiles and Percentiles)
The calculation of mean x and standard
deviation s
What it is –
Descriptive statistics
 Descriptive statistics include the numbers, tables, charts,
and graphs used to describe, organize, summarize, and
present raw data.




central tendency (location) of data, i.e. where data tend to
fall, as measured by the mean, median, and mode.
dispersion (variability) of data, i.e. how spread out data are,
as measured by the variance and its square root, the standard
deviation.
skew (symmetry) of data, i.e. how concentrated data are at the
low or high end of the scale, as measured by the skew index.
kurtosis (peakedness) of data, i.e. how concentrated data are
around a single value, as measured by the kurtosis index.
Pareto Diagram
 Pareto Diagram display orders each type of
failure or defect according to its frequency.
 For a computer-controlled lathe whose
performance was below par, workers
recorded the following
causes and their frequencies:
power fluctuations
6
controller not stable
22
operator error
13
worn tool not replaced 2
other
5
Minitab14
 1. Stat->Quality tools->Pareto chart
 2. Choose chart defects table as
follows
Output
Pareto diagram
 Pareto diagram: depicts Pareto’s
empirical law that any assortment of
events consists of a few major and
many minor elements.
 Typically, two or three elements will
account for more than half of the
total frequency, i.e., it points out the
main causes.
Pareto diagram--application
 Software testing
 Software defect distribution
Count
Design
27%
Other
10%
Requirements
56%
1.0
100
0.8
80
0.6
60
0.4
40
0.2
20
0.0
Soft-defect
Count
Percent
Cum %
Requirement
0.56
56.0
56.0
design
0.27
27.0
83.0
other
0.10
10.0
93.0
Code
0.07
7.0
100.0
0
Percent
Pareto Chart of Soft-defect
Code
7%
Dot diagram
 Second step to improve the quality of lathe,
 Data were collected from observation on the
deviations of cutting speed from the target value set
by the controller.
 EX. Cutting speed – target speed
 3 6 –2 4 7 4
 Dot diagram: A number line in which one dot is placed
above a value on the number line for each occurrence
of that value. That is, one dot means the value
occurred once, three dots mean the value occurred
three times, etc.
 In minitab: stat->dotplots->simple
Dot diagram
 This diagram visually summarize the
information that the lathe is generally
running fast.
Multiple sample
 C1: 0.27 0.35 0.37
 C2: 0.23 0.15 0.25 0.24 0.30 0.33 0.26
Dotplot of C1, C2
0.15
0.18
0.21
0.24
0.27
Data
0.30
0.33
0.36
Variable
C1
C2
Frequency distributions
 A frequency distribution is a
tabular arrangement of data whereby
the data is grouped into different
intervals, and then the number of
observations that belong to each
interval is determined.
 Data that is presented in this manner
are known as grouped data.
Data001.
80 data of emission (in ton)of
sulfur oxides from an industry plant
 15.8 26.4 17.3 11.2 23.9 24.8 18.7 13.9 9.0 13.2
22.7 9.8 6.2 14.7 17.5 26.1 12.8 28.6 17.6 23.7 26.8
 22.7 18.0 20.5 11.0 20.9 15.5 19.4 16.7 10.7 19.1
15.2 22.9 26.6 20.4 21.4 19.2 21.6 16.9 19.0 18.5
23.0
 24.6 20.1 16.2 18.0 7.7 13.5 23.5 14.5 14.4 29.6
19.4 17.0 20.8 24.3 22.5 24.6 18.4 18.1 8.3 21.9
12.3
 22.3 13.3 11.8 19.3 20.0 25.7 31.8 25.9 10.5 15.9
27.5 18.1 17.9 9.4 24.1 20.1 28.5
Class limits & frequnecy
Class limits
5.0 -- 8.9
9.0 – 12.9
13.0 – 16.9
17.0 – 20.9
21.0 – 24.9
25.0 – 28.9
29.0 – 32.9
Total
Frequency
3
10
14
25
17
9
2
80
Class limit and width
 lower class limit: The smallest value that
can belong to a given interval
 upper class limit: The largest value that
can belong to the interval.
 Class width: The difference between the
upper class limit and the lower class limit is
defined to be the class width.
Guidelines for classes






1. There should be between 5 and 20 classes.
2.The class width should be an odd number. This will
guarantee that the class midpoints are integers instead of
decimals.
3. The classes must be mutually exclusive. This means that
no data value can fall into two different classes
4. The classes must be all inclusive or exhaustive. This
means that all data values must be included.
5. The classes must be continuous. There are no gaps in a
frequency distribution. Classes that have no values in them
must be included (unless it's the first or last class which are
dropped).
6.The classes must be equal in width. The exception here is
the first or last class. It is possible to have an "below ..." or
"... and above" class. This is often used with ages
Steps
 1. Find the largest and smallest values
 2. Compute the Range = Maximum Minimum
 3. Select the number of classes desired.
This is usually between 5 and 20.
 4. Find the class width by dividing the
range by the number of classes and
rounding up.
You must round up, not off.
Normally 3.2 would round to
be 3, but in rounding up, it
becomes 4.
Class limits & frequnecy
Class limits
[5.0, 9.0)
[9.0, 13.0)
[13.0, 17.0)
[17.0, 21.0)
[21.0, 25.0)
[25.0, 29.0)
[29.0, 33.0)
Total
Frequency
3
10
14
25
17
9
2
80
Variants of frequency distribution
 The cumulative frequency distribution is
obtained by computing the cumulative
frequency, defined as the total frequency of
all values less than the upper class limit of
a particular interval, for all intervals.
 Relative frequency: the ratio of the number
of observations in the interval to the total
number of observations
 The percentage frequency distribution is
arrived at by multiplying the relative
frequencies of each interval by 100%.
Cumulative frequency
Class limits
Less than 5
Less than 9
Less than 13
Less than 17
Less than 21
Less than 25
Less than 29
Less than 33
Frequency
0
3
13
27
52
69
78
80
Percentage distribution
Class limits
Perc. Dist.
Frequency
[5.0, 9.0)
[9.0, 13.0)
[13.0, 17.0)
[17.0, 21.0)
[21.0, 25.0)
[25.0, 29.0)
[29.0, 33.0)
Total
3.75%
12.5%
17.5%
31.25%
21.25%
11.25%
2.5%
100%
3
10
14
25
17
9
2
80
Histogram
 The most common form of graphical
presentation of a frequency
distribution is the histogram.
 Histogram: is constructed of adjacent
rectangles; the height of the
rectangles is the class frequencies
and the bases of the rectangles
extend between successive class
boundaries.
Histogram in Minitab
Histogram in Minitab
1. Graph->histogram->simple
2. Graph variables: c4 (all data in a column)
3. Edit bars: Click the bars in the output figures, in
Binning, Interval type select midpoint and interval
definition select midpoint/cutpoint, and then input 7
11 15 19 23 27 31 as illustrated in the following
Density histogram
 When a histogram is constructed from a
frequency table having classes of unequal
lengths, the height of each rectangle must
be changed to
 Height = relative frequency / width.
 The area of the rectangle then represents
the relative frequency for the class and the
total area of the histogram is 1.
Density histogram
Density Histogram




Graph->histogram->simple
Scale->Y-Scale Type->Density
Edit Bars->Binning->Cut point->
5 13 17 21 25 29 33
Cumulative histogram
 1) Graph>histogram>simple
 2) Dataview->
Datadisplay: check
“symbos” only
Smoother: check
“lowess” and “0” in
degree of
smoothing and “1”
in number of steps.
Stem-and-leaf Display
 Class limits and frequency, contain data in each class,
but the original data points have been lost.
 Stem-and-leaf: A data plot which uses part
of the data value as the stem and the rest of
the data value (the leaf) to form groups or
classes. This is very useful for sorting data
quickly.
 Stem-and-leaf: function the same as histogram but
save the original data points.
 Example: 11 numbers:
 12, 13, 21, 27, 33, 34, 35, 37, 40, 40, 41
 Frequency table
Class limits
Frequency
10 – 19
2
20 – 29
2
30 – 39
4
40 – 49
3
Stem-and-leaf
Stem-and-leaf: each row has a stem and
each digit on a stem to the right of the vertical
line is a life.
The "stem" is the left-hand column which
contains the tens digits.
The "leaves" are the lists in the right-hand
column, showing all the ones digits for each
of the tens, twenties, thirties, and forties.
Key: “4|0” means 40
Stem-and-leaf Display
 Example in P23: 20 numbers:
 29, 44, 12, 53, 21, 34, 39, 25, 48, 23
 17, 24, 27, 32, 34, 15, 42, 21, 28, 27
Frequency table
Class limits
Frequency
10 – 19
3
20 – 29
9
30 – 39
4
Stem-and-leaf
40 – 49
3
1|257
50 – 59
1
2|113457789
3|244 9
4|248
5|3
Stem-and-leaf in Minitab
 The display has three columns:
 The leaves (right) - Each value in the leaf
column represents a digit from one observation.
 The stem (middle) - The stem value represents
the digit immediately to the left of the leaf digit.
 Counts (left) - If the median value for the
sample is included in a row, the count for that
row is enclosed in parentheses. The values for
rows above and below the median are
cumulative.
Stem-and-leaf for DATA001


Stem-and-leaf of frequencies N = 80
Leaf Unit = 1.0













2 0 67
6 0 8999
11 1 00111
17 1 223333
24 1 4445555
32 1 66677777
(13) 1 8888888999999
35 2 0000000111
25 2 222223333
16 2 4444455
9 2 66667
4 2 889
1 3 1
Ch2.5: Descriptive measures
 Mean: the sum of the observation divided
n
by the sample size.
xi

x  i 1
n
 Median: the center, or location, of a set of
data. If the observations are arranged in an
ascending or descending order:
 If the number of observations is odd, the
median is the middle value.
 If the number of observations is even, the
median is the average of the two middle values.
Example
 15 14 2 27 13
 Mean:
15  14  2  27  13
x
 14.2
5
 Ordering the data from smallest to
largest
 2 13 14 15 27
 The median is the third largest value
14
Other central tendency
 Midrange
 The midrange is simply the midpoint
between the highest and lowest values.
 Mode
 The mode is the most frequent data
value. There may be no mode if no one
value appears more than any other.
There may also be two modes (bimodal),
three modes (trimodal), or more than
three modes (multi-modal).
Summary
 The Mean is used in computing other statistics (such
as the variance) and does not exist for open ended
grouped frequency distributions. It is often not
appropriate for skewed distributions such as salary
information.
 The Median is the center number and is good for
skewed distributions because it is resistant to change.
 The Mode is used to describe the most typical case.
The mode can be used with nominal data whereas the
others can't. The mode may or may not exist and
there may be more than one value for the mode
 The Midrange is not used very often. It is a very
rough estimate of the average and is greatly affected
by extreme values (even more so than the mean).
Summary cont.
Preporty
Mean
Median Mode Midrange
Always
Exists
No
Yes
No
Yes
Uses all
Yes
data values
No
No
No
Affected by Yes
extreme
values
No
No
Yes
Sample variance
 Deviations from the mean:
n
s2 
2
(
x

x
)
 i
i 1
n 1
n
s2 
 Standard deviation s:
n
s
2
(
x

x
)
 i
i 1
n 1
n
n   x  ( xi )2
i 1
2
i
i 1
n(n  1)
Quartiles and Percentiles
 Quartiles: are values in a given set of
observations that divide the data in 4 equal
parts.
 The first quartile, Q1 , is a value that has one
fourth, or 25%, of the observation below its
value.
 The sample 100 p-th percentile is a value
such that at least 100p% of the observation
are at or below this value, and at least
100(1-p)% are at or above this value.
Example
 Example in P34:
14.7  15.2
Q1 
 14.95
2
19.0  19.1
Q2 
 19.05
2
22.9  23
Q3 
 22.95
2
N/4 is an
integer, take
the average;
Or round up,
otherwise
Boxplots
 A boxplot is a way of summarizing
information contained in the quartiles
(or on a interval)
 Box length= interquartile range= Q3  Q1
Quartile calculation in Minitab
 The first quartile (Q1) is the observation at position
(n+1) / 4, and the third quartile (Q3) is the
observation at position 3(n+1) / 4, where n is the
number of observations. If the position is not an
integer, interpolation is used.
 For example, suppose n=10. Then (10 + 1)/4 = 2.75,
and Q1 is between the second and third observations
(call them x2 and x3), three-fourths of the way up.
Thus, Q1 = x2 + 0.75(x3 - x2). Since 3(10 + 1)/4 =
8.25, Q3 = x8 + 0.25(x9 - x8), where x8 and x9 are
the eight and ninth observations.
 Indeed, Choose “Hinges” in BoxEndpoints, will get
Quartile as in Textbook.
Modified boxplot
Upper limit = Q3 + 1.5 (Q3 - Q1)
 Outlier: too far from third
quartile.
 Largest observation
within 1.5(interquartile
range) of third quartile.
 Modified boxplot: identify
outliers and reduce the
effect on the shape of the
boxplot.
Lower limit = Q1- 1.5 (Q3 - Q1)
Homework 1
 Problems in Textbook (2.62,
2.67,2.71, 2.72, 2.75) 4 points
 Due date: next lecture.
Conclusion
 Graph the data as a dot diagram or
histogram or box plot to assess the
overall pattern of data
 Group the data by frequency
distribution, Stem-and-leaf
 Calculate the summary statisticssample mean, standard deviation,
and quartiles – to describe the data
set.
The END
Thanks !