stat recap

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


2/23

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


3/23

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.


4/23

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


5/23

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


6/23

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)


7/23

Four Types of Probability

Marginal Probability

Union Probability

Joint Probability Conditional Probability


8/23

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


9/23

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


10/23

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


11/23

Some Special

Distributions Discrete binomial Poisson

hypergeometric Continuous normal uniform exponential


12/23

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


13/23

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


14/23

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


15/23

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


16/23

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


17/23

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.

.


18/23

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


19/23

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


20/23

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


21/23

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


22/23

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


23/23

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

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