Stat Recap
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Transcript of Stat Recap
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7/31/2019 Stat Recap
1/23
Recap What we did so far
Lecture Class Relevance of Statistics Data : representation Measures of central tendency
and variability Probability Laws Chebysheve theorem Discrete distributions
Uniform Binomial Poisson
Geometric Hypergeometric etc.
Lab Class
Lab 1: DataRepresentation
Lab2: Examples on
Probability
Lab 3: Binomial &Poisson
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7/31/2019 Stat Recap
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Recap What we did so far
Lecture Class Continuous Distributions
Uniform Normal
Exponential Gamma Erlang etc.
MGF and its properties
Central Limit Theorem
Lab 4:Normal andother
Lab 5: MGF and otherproperties
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7/31/2019 Stat Recap
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What is Statistics?
Science of gathering, analyzing,interpreting, and presenting data
Branch of mathematics
One page in Courses of study?
Facts and figures
Measurement taken on a sample
Type of distribution being used to analyze
dataStatistics is the scientific method thatenables us to make decisions as
responsibly as possible.
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Statistics The science of data to answerresearch questions
Formulate a research question(s)(hypothesis) Collect data Analyze and summarize data
Draw conclusions to answer researchquestions Statistical Inference
In the presence of variation
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Common Statistical Graphs Histogram -- vertical bar chart of
frequencies Frequency Polygon -- line graph of
frequencies
Ogive -- line graph of cumulativefrequencies
Pie Chart -- proportional representation
for categories of a whole Stem and Leaf Plot
Pareto Chart
Scatter Plot
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Methods of Assigning
Probabilities Classical method of assigning
probability (rules and laws) Relative frequency of occurrence
(cumulated historical data) Subjective Probability (personal
intuition or reasoning)
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Four Types of Probability
Marginal Probability
Union Probability
Joint Probability Conditional Probability
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Measures of Central Tendency
Measures of central tendency yieldinformation about particular places
or locations in a group of numbers. Common Measures of Location
Mode
Median Mean
Percentiles
Quartiles
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Measures of Variability
Measures of variability describe thespread or the dispersion of a set ofdata.
Common Measures of Variability Range
Interquartile Range
Mean Absolute Deviation Variance
Standard Deviation
Z scores and Coefficient of Variation
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Probability
Distributions.. Two Types of Probability Distributions Continuous When a variable being measured is
expressed on a continuous scale, its probabilitydistribution is called a continuous distribution. Theprobability distribution of piston-ring diameter iscontinuous.
Elapsed time between arrivals of bank customers Percent of the labor force that is unemployed
Discrete When the parameter being measured can only
take on certain values, such as the integers 0, 1, 2, ,the probability distribution is called a discretedistribution. The distribution of the number ofnonconformities would be a discrete distribution.
Number of new subscribers to a magazine Number of bad checks received by a restaurant
Number of absent employees on a given day
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Some Special
Distributions Discrete binomial Poisson
hypergeometric Continuous normal uniform exponential
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The Expected Value ofX
LetXbe a discrete rv with set of
possible valuesD and pmfp(x). The
expected value or mean value ofX,denoted
( ) ( )X x D
E X x p x
( ) or , isXE X
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The Variance and Standard
DeviationLetXhave pmfp(x), and expected value
Then the variance ofX, denoted V(X)
2 2(or or ), isX
2 2( ) ( ) ( ) [( ) ]
D
V X x p x E X
The standard deviation (SD) ofXis
2X X
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7/31/2019 Stat Recap
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Binomial Distribution
Probabilityfunction
Meanvalue
Varianceandstandard
deviation
P X
n
X n XX n
X n X
p q( )!
! !
for 0
n p
2
2
n p q
n p q
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Poisson Distribution
Probability function
P X
X
X
where
long run average
e
X
e( )!
, , , ,...
:
. ...
for
(the base of natural logarithms)
0 1 2 3
2 718282
Mean value
Standard deviation Variance
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Hypergeometric Distribution
Probability function Nis population size
nis sample size
Ais number of successes in
population xis number of successes insample
A n
N
2
2
2
1
A N A n N n
NN
( ) ( )
( )
P xC C
C
A x N A n x
N n
( )
Mean
value
Variance and standard deviation
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Uniform DistributionMean and Standard Deviation
Mean
=+
a b
2
Mean
=+
41 47
2
88
244
Standard Deviation
b a12
Standard Deviation
47 4112
63 464
1 732.
.
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Characteristics of theNormal Distribution
Continuousdistribution
Symmetricaldistribution
Asymptotic to thehorizontal axis Unimodal A family of curves Area under the curve
sums to 1. Area to right of meanis 1/2.
Area to left of meanis 1/2.
1/2 1/2
X
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Exponential Distribution
Continuous
Family of distributions
Skewed to the right Xvaries from 0 to infinity
Apex is always at X = 0
Steadily decreases as Xgets larger
Probability functionf X XX
e( ) ,
for 0 0
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7/31/2019 Stat Recap
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Moments and Moment-Generating Functions
The moment-generating function (MGF) of the random
variable Xis given by E(etX) and denoted by Mx(t). Hence
Let X be random variable with MGF Mx(t). Then
)()(tx
X eEtM
x
tx xfe )(
dxxfe
tx)(
0
)(
t
r
x
r
rdt
tMd
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MGF..(a) If X is a discrete r.v., the
.(b) If X is a continuous r.v., then
.
)x(pe)t(M
x
xt
X
dx)x(fe)t(Mxt
X
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Central Limit Theorem
Consider a set of independent, identically distributedrandom variables Y1... Yn, all governed by an arbitrary
probability distribution with mean and finite variance
2. Define the sample mean,
n
i
inYY
1
1
Central Limit Theorem. As n , thedistribution governing approaches a Normal
distribution, with mean and variance 2/nY
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Coverage up to Minor I General concepts about datarepresentation and use of statistics
Probability laws and Its application
including Chebyshevs Theorem, BayesTheorem etc. Various Discrete distributions Various Continuous Distributions
MGF and its properties Central Limit Theorem