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Economics

ECON 2B03

Jeff Racine

Fall

Description

Overview 9/11/2013 5:37:00 AM
Statistics: is a branch of mathematics dealing with the collection,
presentation, and interpretation of data
The field can be divided into two broad categories:
Descriptive statistics: which seeks to describe general
characteristics of a set of data
Inferential statistics: which seeks to draw inferences about
(unknown) features of a population based on a (known) sample
drawn from a population
Broadly speaking, statistics provide a rigors framework for describing,
analyzing, and drawing on the basis of data
Why do we need statistics?
Mainly due to uncertainty (we cannot know with certainity many
things in life, even though they clearly exist)
Variables and Distributions
In any statistical study, we analyze info/data obtained from
individuals
Individuals can be people, animals, households, plants, and or
objects of interest
A variable is any characteristic of an individual
A variable varies among individuals (as opposed to constants)
o Ex. Age, income
The distribution of a variable tells us what values the variables
takes and how often/likely it takes these values
Population and Samples
Population: set of all possible observations on some characteristic
(eg. Age, height, income)
Sample: subset of a population
Random sample: one obtained if every member of the population
has an equal of being in the sample
Categorical population: a population whose characteristic is
inherently nonnumerical (e.g. sex, race) Quantitative population: a population whose characteristic is
inherently numerical
o For categorical population , we are typically interested in
proportions (e.g. the proportion who are male)
o For quantitative population, we are typically interested in
proportions (e.g. the average, and how „spread-out‟ the
variable is)
Types of Categorical Data
Nominal data
o Numbers merely label difference in kind
o Cannot be manipulated arithmetically
o Can only be counted
o Ex. Indicator for a categorical variable (o=male, 1=female)
Ordinal data
o Label difference in kind
o Order or rank observation based on importance
o No meaningful arithmetic analysis possible
o Can only be counted and ranked
o Ex. Ranking firms according to profit (e.g. HSBC) 9/11/2013 5:37:00 AM
Presenting Data: Tables and Graphs
When presenting data either in the form of tables or graphs, it is
often desirable of first split the data into groups/classes
How we split the data into groups/clases
o Data types:
Categorical (nominal or ordinal)
Quantitative (discrete or continuous)
o Regardless of the data type, data classes should be:
Collectively exhaustive (must exhaust all logical
possibilities for classifying available data)
Mutually exclusive (must not overlap or have data in
common)
One immediately confronts the issue of how many classes to create
o Desirable class number:
Should fit data type
Often recommended: between 5 to 20
Sturgess‟s rule: desirable number of classes=K, an
integer, where k is the integer closest to (use standard
rules for rounding)
1+3.3log n10
where n is the sample size, and where log n 10 the
power to which the base (10) is raised to yield n
ex. If n=1000 then log 1100=3, so 1+3.3
log10000=10.9, and therefore k=11 so by rule
use 11 classes when n=1000
Data Classes
Class width= the difference between the lower and upper limits of a
class
To achieve uniform class widths in a table, divide data set width by
desirable class number
Approximate class width:
o Large value= small value/ desirable class number
Tabular Components
We first consider creating effective tabular summaries
An effective table includes: o Number (often based on chapter or page numbers)
o Title (focus on what, where and when)
o Caption (brief verbal summary)
o Footnotes (e.g. size of sampling error, likely extent of
systematic error, etc)
o Decimals (consistent number of decimals)
o Rounding (consistent rounding rules)
o Class sums (sum of data pertaining to each class, crucial for
open- ended classes)
Frequency distributions
Frequency distribution are an effective way to summarize data
Absolute frequency distributions
o Absolute class frequency:
Absolute number of observation that fall into a given
class
o Absolute frequency distribution:
Tabular summary of data set
Shows absolute numbers of observations that fall into
several data classes
o In R there is a function list() used for graphical
representation
Relative Frequency distributions:
Relative class frequency:
o Ratio of a particular class‟s number of observation to the total
number of observations made
Relative frequency distributions:
o Tabular summary of data set
o Shows portions of all observations that fall into each of
several data classes
Cumulative Frequency Distributions
Cumulative class frequency
o The sum of all class frequency up to and including the class in
question LE distribution (typical, often called „cumulative; distribution)
LE: less then or equal to upper class limit
Moves from lesser to greater classes
ME distribution (less used, often classed „survivors‟ distribution)
ME: more then or equal to lower class limit
Moves from greater to lesser classes
Cross Tabulations
Cross tabulation are tabular summaries for two variables
One variable represented by row heads
The other variable represented by column heads
Information for both variables entered in table cells
In R we use the xtabs() functions
Sept.10.2013
Graph Types
Core Graphs
o Two dimensional graphs
o Ex. Histograms, scatter, diagrams, etc.
Specialty graphs
o Combine elements from core graphs to display data in unique
ways
o Ex. Bar graphs, pie, charts. Etc.
3G graphs
o display 3 elements; width, height, depth
Frequency Histograms
Horizontal axis
o Identifies upper and lower limits of data classes
Vertical axis
o Shows number of observations in each class (absolute freq)
o Shows ratio of class freq to class width (relative freq)
Rectangles o Represents corresponding class frequencies by area or by
height (if all class intervals are alike)
In R we use the hist() function (hist(.prob=TRUE) for relative freq
divided by the class width
Frequency Polygon and Density Estimations
Frequency polygons
o Portray frequency distributions as many-sided figures
o Class mark= avg of two class limits
o Final points on horizontal axis lie ½ length below the lowest
(or above the highest) class limit
Density estimator
o Portray freq distributions as smooth curves
o Remove irregularities from histograms depicting info gathered
is sample surveys
o Estimate how histograms would appear in census info were
graphed with many tiny classes
o In R we use the plot() function
Ways to chart categorical data
Unlike quantitative, when a variable is categorical the data in the
graph can be ordered any way we want (alphabetical, by increasing
value, by personal prefence)
Bar Graph:
o Each category is represented by a bar
o In R, see the barplot() function
Pie charts:
o The slices must represent the pieces
o In R pie()
Summary Statistics: Symbolic Expression
Population parameters (typically unknown)
o Summary statistics based on population data
o Designated by Greek letter (eg. Mu, sigma)
Sample Statistics (computed from sample)
o Summary statistics based on sample data o Designated by roman letter (eg. X(with bar above),s)
Observed values
o Traditionally represented by x or X
o Different value indicated by subscripts 1,2,3,…etc.
Eg. X1, X2
Observation totals
o Population total= N
o Sample total=n
Sept. 12.2013
Summary Statistics: Summation notation
In statistics, we often need to accumulate or sum some variable of
interest
For example, A might denote, say, income, with A1 being the
income of the first person in our sample, A2 that for the second and
son on
It is supremely convenient to avoid having to write things like
o A1+A2+A3+…….OR
o (A1-B)+(A2-B)+(A3-B)+…… or even
o (A1-B)^2+(A2-B)^2+(A3-B)^2+……
One convenient operator that we shall use heavily is the summation
operator
In R use the sum() function
The sum of a variable X can be written as
o Summation(from i=1 to n) Xi= X1+X2+….+Xn
o Note Summation(from i=1 to n) and Summation (from i*)
and Summation are often interchanged
o Care must be taken when expanding sums. For instance
2 2 n
Summation(from i=1 to n) X i=X
square the summation before and after are differnetnt
Some useful manipulations involving the summation operator follow
from the definition of the arithmetic mean,
o X(BAR) = 1/n Summation(from i=1 to n)Xi o Common ones you may find useful are as follows:
Summation(from i=1 to n)Xi= nX(bar)
Summation(from i=1 to n)(Xi- Xbar) =0
Summation(from i=1 to n)(Xi-Xbar)^2=
Summation(from i=1 to n) X i-nXbar^2
Summary Measure Types
Central Tendency Measures
o Locate a data set‟s „center‟
Dispersion measure
o Focus on the spread of data around a set‟s center
Shape Measure
o Describe the symmetry (or asymmetry) of a
frequency/density curve, as well as how flat or peaked it is
Proportions
o Used for categorical data
o Describe the frequency of observations in a category as
fraction of all observations made
Central Tendency Measures
The Arithmetic Mean
o Point of balance
o Beware- the mean can be strongly affected by the presence
of „extreme‟ value (potential „outliers‟)
o Calculation let Xi, denote an observation , i=1,2,….
Arithmetic Mean:
Population: mu= Summation(from i=1 to n)Xi/N
Sample: Xbar= Summation(from i=1 to n)Xi/n
o In R we use the mean() function
The Median
o Divides data into equal halves
Odd number of observations:
Median= middle observation in the ordered array
Even number of observations: Median- average of the two middle in the ordered
array
o Calculation: After sorting your data in ascending order
Median (n odd) median=X (n+1)/2
Median( n even) median= (X n/2+ X (n/2)+1)2
The mode: the most frequently occurring value in a data set
o Can be found either extreme of data set (and thus be atypical
of the majority of observations)
o May not appear in a data set at all (because no single
observation occurs more than once)
o May occur more than once in a data set
Multimodal frequency distribution: data set has 2 more
or more values with the highest freq (eg. A data set
with 2 modes is termed „bimodal;)
o The mode could also be obtained from a density curve, and is
that values(s) of X lying directly below the peak(s) of the
density curve
Dispersion Measure
The Variance (Average (Squared) Deviation)
o Most common measure of dispersion
o Measure the typical squared deviation about the mean
o Calculated by averaging the squares of individual deviation
from the mean
Calculation
Variance:
o Population: mu = (Summation(from i=1 to n)(Xi- mu)^2)/N
2
o Sample: s =(Summation(from i=1 to n) (Xi- Xbar)^2)/n-1
In R we use the var() function (note this computes the sample
variance)
Standard Deviation
The positive square root of the variance
Falls in the same range of magnitude (and appears in the same
units) as the observation themselves Calculation:
o Population: sigma= square root(sigma^2)
o Sample: square root (s^2)
o In R we use the sd() function
Dispersion Measures
Overall Range
o Equals the difference between the largest and smallest
observation in a data set( in R we could use the range()
function)
range(X)
Interfractile Ranges
o Measures difference between 2 valus in the ordered array
(called „fractiles‟)
Quartiles: divide the array into 4 quarters
Interquartile Range: difference between 3 rdand 1 st
quartiles (contain middle 50% of data)
In R we use the IQR() and quantile() functions
Shape Measures: Skewnes
A frequency distribution‟s degree of distortions from horizontal
symmetry
Pearson‟s (first) coefficient of skew is (we want to know is it +/- ?):
o Skewness= mean-mode/ standard deviation
o An alternative (more popular, default in R) moment-based
measure is given by
Skewness= Summation (i=1 to n)( (X1-Xbar)^3)/n/(
Summation (i=1 to n)( (X1-Xbar)^2)/n)^3/2
o Skewness=0 fro symmetric distributions
o For right-skewed distributions the mean is bigger than the
median which is bigger than the mode (for left-skewed just
the opposite holds)
Shape Measures: Kurtosis
A frequency curve‟s degee of peakness Coefficient of kurtosis
Kurtosis= = Summation (i=1 to n)( (X1-Xbar)^4)/n / ( Summation
(i=1 to n)( (X1-Xbar)^2)/n)^2
Kurtosis= 3 for the normal distribution that we introduce shortly(
handy reference distribution)
In order to compute certain higher-order “moments” (e.g. skewness
and kurtosis) of a quantitive variable we need to install an optional
R PACKAGE
In RStudio we first install the moments package via the Install
packages icon in the packages tab (lower right pane by default)
The function kurtosis() and skewness() can then be accessed after
we oad the moments package via library (moments)
The five-Number Summmary
The five-number summary of set of observations consists of the
smallest observation, the first quartile, the median, the third
quartile, and the largest observation, written on order from smallest
to largest Minimum Q1 Median Q3 Maximum
In R the function is fivenum()
Suspected Outliers
Outliers are troublesome data points, and it is important to be able
to identify them
One way to raise the flag for suspected outlier is to compare the
distance from the suspicious data point to the nearest quartile (Q1
or Q3)
We then compare this distance to the interquartile range (distance
between Q1 and Q3)
We call an observation a suspected outlier if it falls more than 1.5
times the size of the interquartile range (IQR) below the first
quartile or above the third quartile
This is called the “1.5 IQR rule for outliers”
Boxplots A box-and-whisker plot (sometimes called simply a „box plot‟) is a
histogram-like method of displaying data, invented by John Tukey
o Draw a box with ends at the quartile Q1 and Q3
o Draw the median as a horizontal line in the box
o Now extend the “whiskers” to the farthest points that are not
suspected to outliers (i.e. that are within 1.5 times the
interquartile range (Q3-Q1) from the then of the box)
o Then, for every point more than 1.5 times the interquartile
range from the end of a box, draw a dot
In R the function is boxplot()
Normal („Gaussian‟) Frequency distribution
The normal distribution is a special type of population whose
relative frequency distribution is characterized by:
o Single peak with mean, median, and mode coinciding at the
center of distribution
o Perfect symmetry indefinitely in both directions from center
(approaching, but never touching, the horizontal axis)
o Predictable percentages of all population values the lie within
+/-1, +/-2 and +/-3 standard deviation from the mean
(roughly, 68%, 95%, and 100%)
Chebyshev‟s Theorem
Chebyshev’s Theorem is an empirical rule that applies to all
distributions, not only the Normal distribution
Regardless of he shape of a population frequency distribution, the
proportion of observation falling within k standard devivations of
the mean is at least (given that k quals 1 or more):
o Proportional falling within +/- kσ of μ>=1 - 1/k^2
Coefficient of Variation
o Used to compare degree of dispersion among data sets
o Ratio of the standard deviation to the arithmetic mean
o In R ad()/mean() 9/11/2013 5:37:00 AM
Origins of Data
Data can originate in a number of ways
Internal Data
o Created as by-products of regular activities
o Ex. Customer, employee, production records, government
records
External Data (typical source fro this course)
o Created by entities other than the person, firm, or
government that wants to use the data
o Ex. Print sources, CD-ROMs, web sites
In the past, users were limited to print data (e,,g, print and
CDROM)
o Frequently expensive
o Useful for transferring to computer applications
The internet is an exceptionally efficient source of data
Important Data Websites
Canadian government SOURCES
Foreign Government sources
Fortune 500 company sources
Sampling Versus Census Taking
A census is a complete survey of every member in the population
A sample is a partial survey in which data is collected for only a
subset of the population
Reasons for sampling versus conduction a census
o Expense can be prohibitive
o Speed of response
o Impossible. For example we may have infinite population (I,e,
observation occur from an infinite or recurring process)
o Destructive sampling (e.g., lifetime of light bulb, safety of
automobile)
o Accuracy can, oddly enough, be better (e.g. higher quality
information results via hiring fewer persons who can be better
trained, say) Types of Samples: Non probability
Nonprobability sample: occurs when a sample is taken from an
existing population in a haphazard fashion without the use of some
randomizing device assigning each member a known (positive)
probability of selection
o Voluntary response sample (e.g., phone survey (self
selection issue)
o Convenience sample (eg. Most easily selected persons)
o Judgment sample ( based on data collects personal judgment
(experience) )
o All of the above will likely not be representative of the
population as a whole, and are perhaps the worst types of
samples to model
Types of Samples: probability
Probability sample: occurs when a sample is taken with the help
of a randomizing device that assures each member a known
(positive, not necessarily equal( probability of selection
o Simple random a sample: obtained if every member of the
population has an equal chance of being in the sample (main
type)
st
o Systematic random sample: randomly select 1 element,
then include every kth element thereafter till sample
complete
o Stratified random sample: take random samples from
every stratum (clearly distinguishable subgroups) in a
population
o Clustered random sample: population naturally subdivided
into geographically distinct clusters, and samples are created
by taking censuses in randomly chosen clusters
Errors in Survey Data
We expect that there will exist random error when we conduct a
survey (also known as chance error or sampling error)
Arises only in sample survey (I,e, this will be zero in a census)
Equals difference between o Value obtained from single random sample
o Value obtained by taking a census
Size of error (and frequency) often estimated and reported with
observed data (e.g., “A census would reveal a number within 2% of
the survey value”)
Bias in Survey Data
Sadly, random errors are the least of our problems, and even if we
had a census, we might still encounter serious problems
Systematic errors( a.k.a, bias, nonsampling error) equals the
difference between)
o Value obtained by taking a census
o The (unknown) rue value)
Can be hard to detect
Size of error cannot be estimated
Bias can creep into a survey in a number of ways
o Can be built into the survey‟s design (pre survey)
o Can occur during the survey‟s execution (during survey)
o Can transpire during final procession (post survey)
Selection bias:
Selection bias: systematic tendency to include elementary units
with particular characteristics (while excluding those units with
other (opposite) characteristic)
Source of selection bias
o Nonrandom samples
(e.g. surveying all employees who leave work at 5pm.
o Faulty design of random samples
E.g. selecting every 12th month to survey monthly stock
prices
o Faulty execution of perfect sampling plans
E.g. interviewers substituting other persons fro those
randomly selected)
Nonresponse bias: Systematic tendency for elementary units with particular
characteristics not to contribute data( while other units do)
Sources of Nonresponse Bias
o Contact problems
(e.g., potential respondent tears up mailed
questionnaire, or won‟t answer calls of telephone
surveyor, or refuses to cooperate with surveyor at the
door
o Badly designed questionaires
Eg. Unattractive design; hard-toread print. Questions
that are borinh, intrusive, unclearm excessive in
number, badly sequenced. Multiple-choce questions that
over lap
Response Bias:
Tendency for answers to survey questions to be wrong in some
systematic way
Sources for response bias
o Failing to
Pretest questionnaires
Motivate truthful answers
Define terms clearly
o Asking leading questions
o Inappropriate behavior of interviews and/or respondents
o Processing errors
Surveys versus Experiments
There are addition shortcomings of surveys (a.k.a “observational
studies”)
o Can only reveal association (correlation)
o Data collected by merely recording information about selected
characteristics (while ignoring other characteristics)
One recent development in economics is the use of experimental
data to circumvent the shortcomings associated with observational
studies
Experiments have a number of useful features including: o Can establish causation
o Data is collected by exposing elementary units to some kind
of change or treatment (while leaving all else unchanged)
Association versus Causation
Association
o Can show that A varies with b, but cannot prove the A causes
b
o May or may not indicate causation
Ex. Opening umberallas us associated with
rainfall..which causes which? The answer cannot be
settled purely on the basis of the data, rather ot must
come from outside the data.
Causation
o A has a casual relationship with B(i.e. if A then B)
o Can be established (or refuted) by well-designed experiment
Ex. 1. Have participants randomly open and close
umberalla, observe frequency of rainfall
Experimental limits
With a well designed experiment, we introduce a change into a
carefully controlled setting then observe the consequences, if any:
if they do occur, we know the cause because we introduce it
However, experimentation is not feasible if:
o Physically impossible
Ex. Resolution of issue requires sex or race be assigned
to otherwise identical groups
o Practically impossible
E.g. execution requires participation of large numbers of
people unwilling to cooperate
o Ethically unacceptable
E,g, creating a control group by forbidding head start
enrollment for some disadvantaged children theory of probability 9/11/2013 5:37:00 AM
Basic Probability Concepts
We now lay the foundation for the filed of statistical inference by
studying the theory of probability
Statistical inference
The set of techniques used to turn sample evidence into valid
conclusions about statistical population of interest
The theory of Probability
Calculus of the likelihood of specific occurrences
Gives rise to entire field of inferential statistics
Points to likely truth when we cannot know it with certainty
Basic Probability Concepts
Experiments: any repeatable process from which an outcome
measurement, or result is obtained
The Random Experiment
o An experiment whose outcome cannot be predicted with
certainty
o Any activity that results in one and only one of several clearly
defined possible outcomes, but that does not allow us to tell
in advance which of these will prevail in any particular
instance
The Sample Space
o A listing of all the basic outcomes of random experiment, also
known sometimes as the
Outcome space
Probability space
EX. Tossing a single coin , rolling a single die
The Nature of Random Events
A random Event
o Any subset of the sample space
A simple Event
o Any single basic outcome from a random experiment
o Univariate, bivariate, multivariate
A composite Event o Any combination of two or more basic outcomes
How Random Events Relate
Mutually Exclusive Events
o Random events that have no outcome in common
o Often also called disjoint or incompatible events
Collectively Exhaustive Events
o Random events that contain all basic outcomes in the sample
space
o When the appropriate random experiment is conducted, one
of these events is bound to occur
Complementary Events
o Two random events for which all basic outcomes not
contained in one event are contained in the other event
(denoted A and Ā, the bar denoting „complemetnary‟)
o Such events ar noth mutally exclusive and collectively
exhaustive at the same time
Union of Events (Logical operator U (or))
o All basic outcome contained in one or the other event
Interesection of Events (logical operation [and])
o All basic outcomes contained in one and the other event
___________
General Addition Law for Events which are not Mutually Exclusive
If there are two events, A and B, which are not mutually exclusive, then:
P(A or B) = P(A) + P(B) – P(A and B)
Law of probability
Unconditional probability p(A)
o The likelihood that a particular event will occur, regardless of
whether another event occurs
Joint Probability p(A inter B)
o The likehood that two or more vents will occur simultaneously Conditional probability
o The likelihood that a particular event will occur, given the fact
that another event has already occurred or is certain to occur
Unconditional from Joint Probability Rule
o To obtain an unconditional probability from joint probability,
we sum the joint probability over all possible occurrences of
the other event(s)
o p(A)= Sum p(A intersection B)
ex. Sum o (face inter suit)
=p(facr inter heart) + p(face inter club) + p(face
inter spade) + p(face inter diamond)
=3/52 +3/52 +3/52 +3/52 =12/52
Joint Probability Tables
a joint probability table shows frequencies or relative frequencies
for joint events
passed failed Marginal
prob
Male 3800 4700 8500
Female 1600 2400 400
Marginal total=12500
prob
Joint prob passed failed Marginal prob
Male 0.304 0.376 0.680
Female 0.128
Marginal prob 1
o if we divide each entry by the total number of students we
obtain a se of relative frequencies that can be used to
approximate the probability of occurrence of the relevant
outcome, which we present on the next slide
Laws of Probability: Multiplication the general multiplication law (probability of A and B)
o the joint probability of two events happening at the same
time equals the unconditional probability of one event times
the conditional probability of the other event, given that the
first event has already occurred
p(A inter B) = p(A) * p(B|A)= p(B) * p(A|B)
the events A and B are dependent when the probability of
occurrence of A is affected by the occurrence of B, hence p(A) is not
equal to p(A|B). Independent events are those for which
p(A)=p(A|B)
o the special multiplication law (multiplication law for
independent events)
p(A inter B) =p(A) * p(B)
Sept.26.2013
Bayes’ Theorem (Reverend Thomas Bayes [1702-1761])
How do we estimate or revise probabilities when new information is
obtained? By using Bayes‟ Theorem!
Consider a random experiment having several possible mutually
exclusive outcomes E1…..En. Suppose that the probabilities of these
outcomes, p(E1) have been obtained
We call these Prior Probabilities since they are obtained before
new information is taken into account
A revised probability is called a Posterior Probability
Posterior probability are conditional probabilities, the conditioning
event being new information
By using Bayes‟ Theorem, a prior probability is modified
The new probability is a posterior (conditional) probability
Let A be the new information. By the definition of conditional
probability, the posterior probability of the outcome Ei, is given by
o P(Ei|A)= p(Ei inter A)/ p(A)
Recall from the general multiplicative rule of probability that o P(Ei inter A) = p(Ei) * p(A|Ei)
Substituting (2)into (1)
Background: Discrete Probability Distributions
A probability distribution is the theoretical (i.e., analytical)
counterpart of the frequency distribution discussed earlier
By applying the laws of probability we can determine the probability
that a variable takes on a specific value or range of values
We can always convert a categorical variable into a quantitative one
by assigning it a numerical value, and it is often more convenient
notationally speaking to do so
o The simplest example is a male/female sample where males
are assigned the value 1 and females the value 0
o In this case we talk about random variable X denoting a
person‟s sex with XE{0,1}
By way of example, consider tossing a fair coin 3 times. There are 8
possible outcomes, each outcome is equally likely, and has a
probability 1/8
The outcomes are a1=(h,h,h), a2=(h,h,t) a3=(h,t,h), a4=(t,h,h)
a5=(h,t,t) a6=(t,h,t) a7-(t,t,h) a8=(t,t,t)
Letting X denote the number of heads obtained in 3 tosses, find
p(X=2)
We see that p(X-2)=p(a2)_p(a3)+p(a4)= 3/8
Basic Concepts
Random variable: any quantitative variable the numerical value of
which is determined by a random experiment, and thus by chance
Probability distribution: a table, graph, or formula that
associates each possible value, x, of a random variable, X, with its
probability of occurrence, p(X=x)
Summary measures for the probability distribution are often
desired, such as the mean and standard deviation
If the variable is a discrete random variable, the distribution is a
discrete probability distribution and can be represented in:
o Tabular form o Graphical form
o Symbolic form (e.g, p(X=x))
No probability can be negative (p(X=x) >=0)
The sum of the probabilities of all values of the discrete random
variable must equal 1
Cumulative distribution function: Let X be discrete random
variable and let x be any real number, The CDF of X is the function
F(x)=p(X<=x)
F(x) denotes the probability that the random variable X assumes a
value less than or equal to x
o For ant value b,0 <=F(b) <=1
o If a0.5
Let X be the number of successes in n trials. Then Y=n-X is the
number of failures in n trials. For example, if n=10 then for any
specific Binomial Summary measures
For a binomially distributed random variable, we can obtain the
following results:
o Arithmetic Mean or Expected value:
o variance
o Standard Deviation
Normal Approximation to the Binomial
What is the shape of the binomial distribution? If pi=0.5, then the
binomial distribution is symmetric
It can be shown, by means of the central limit theorem, that a bell
shaped approximation works fairly well provided that n*pi>=5 and
n(1-pi)>=5
Because of this the normal distribution is sometimes used to
approximate the binominal distribution due to its ease of use
The Sample Proportion of Success P=X/n
Often we are interested in the proportion of successes in a sample
rather than the number of successes
Recall that the sample proportion is denoted as P(hat)= X/n. The
random variable must take on the values 0,1…n, so the sample
proportion must take on the value 0/n,1/n..1
It follows that the probability for X=3 is the same as the probability
for P(har)=3/n
Therefore, the probability distribution for the random variable
P(hat) can be derived from the probability distribution for the
random variable X
The mean of P(hat) is pi, and the variance is pi(1-pi)/n
Continuous Probability Density Functions
Continuous random variables
o Unlike their discrete counterparts, continuous random
variables are „uncountable‟ and can assume any value on the
real number line o Rather than looking at p(X=x) (i.e the probability that a
continuous random variable lies in an interval)
The probability density function
o Commonly shows probabilities with ranges of values along the
continuum of possible values that the random variable might
take on
o Let some smooth curve how the probability of a continuous
random variable X is distributed. If the smooth curve can be
represented by a formula f(x), then the function f(x) is called
the probability density function
Areas will be seen to play an important role in determining
probabilities
Since there exist an infinite number of points in an interval, we
cannot assign a non-zero probability to each point and still have
their probabilities sum to one, therefore, the probability of any
specific occurrence of a continuous random variable is defined to be
zero
The total are under the probability density function (just as under
the relative frequency histogram) must therefore equal 1
We shall study two common distributions
o The uniform probability distribution
o The normal probability distribution
Basic Rules for probability density functions
If a smooth curve is to represent a probability density function,
then the following two requirements must be met:
o The total area between the curve and the horizontal axis must
be unity i.e. Integral (-infinity to +infinity)F(x)dx=1
o The curve must never fall below the horizontal axis i.e,
f(x)>=0
Note that the probability of obtaining any one value is zero
The probability that X falls between 2 values a and b, p(a

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