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STAT 552PROBABILITY AND

STATISTICS II

INTRODUCTIONShort review of S551

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WHAT IS STATISTICS?

• Statistics is a science of collecting data,

organizing and describing it and drawing

conclusions from it. That is, statistics is

a way to get information from data. It is

the science of uncertainty.

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BASIC DEFINITIONS

• POPULATION: The collection of all items of interest in a particular study.

•VARIABLE: A characteristic of interest about each

element of a population or sample.

•STATISTIC: A descriptive measure of a sample

•SAMPLE: A set of data drawn from the population;

a subset of the population available for observation

•PARAMETER: A descriptive measure of the

population, e.g., mean

STATISTIC

• Statistic (or estimator) is any function of a r.v. of r.s. which do not contain any unknown quantity. E.g.o are statistics.

o are NOT.

• Any observed or particular value of an estimator is an estimate.

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)X(xam),X(nim,n/X,X,X ii

ii

n

1ii

n

1i

n

1iii

n

1ii

n

1ii /X,X

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• The set of all possible outcomes of an experiment is called a sample space and denoted by S.

• Determining the outcomes.– Build an exhaustive list of all possible

outcomes.– Make sure the listed outcomes are mutually

exclusive.

Sample Space

RANDOM VARIABLES• Variables whose observed value is determined

by chance• A r.v. is a function defined on the sample space

S that associates a real number with each outcome in S.

• Rvs are denoted by uppercase letters, and their observed values by lowercase letters.

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DESCRIPTIVE STATISTICS

• Descriptive statistics involves the arrangement, summary, and presentation of data, to enable meaningful interpretation, and to support decision making.

• Descriptive statistics methods make use of– graphical techniques– numerical descriptive measures.

Types of data – examplesExamples of types of data

Quantitative

Continuous Discrete

Blood pressure, height, weight, age

Number of childrenNumber of attacks of asthma per week

Categorical (Qualitative)

Ordinal (Ordered categories) Nominal (Unordered categories)

Grade of breast cancerBetter, same, worseDisagree, neutral, agree

Sex (Male/female)Alive or deadBlood group O, A, B, AB

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POPULATION SAMPLE

PROBABILITY

STATISTICAL INFERENCE

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• PROBABILITY: A numerical value

expressing the degree of uncertainty regarding the occurrence of an event. A measure of uncertainty.

• STATISTICAL INFERENCE: The science of drawing inferences about the population based only on a part of the population, sample.

Probability

P : S [0,1]

Probability domain range

function

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THE CALCULUS OF PROBABILITIES

• If P is a probability function and A is any

set, then

a. P()=0

b. P(A) 1

c. P(AC)=1 P(A)

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ODDS• The odds of an event A is defined by

( ) ( )( ) 1 ( )C

P A P AP A P A

•It tells us how much more likely to see the occurrence of event A.

ODDS RATIO

• OR is the ratio of two odds.

• Useful for comparing the odds under two different conditions or for two different groups, e.g. odds for males versus females.

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CONDITIONAL PROBABILITY

• (Marginal) Probability: P(A): How likely is it that an event A will occur when an experiment is performed?

• Conditional Probability: P(A|B): How will the probability of event A be affected by the knowledge of the occurrence or nonoccurrence of event B?

• If two events are independent, then P(A|B)=P(A)

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CONDITIONAL PROBABILITY

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1)|(0

0)()(

)(B)|P(A

BAP

BPifBP

BAP

)|()()|()()( BAPBPABPAPABP

),...,|()...,|()|()()...( 1121312121 nnn AAAPAAAPAAPAPAAAP

BAYES THEOREM

• Suppose you have P(B|A), but need P(A|B).

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0)B(Pfor)B(P

)A(P)A|B(P

)B(P

)BA(P)B|A(P

Independence• A and B are independent iff

– P(A|B)=P(A) or P(B|A)=P(B)– P(AB)=P(A)P(B)

• A1, A2, …, An are mutually independent iff

for every subset j of {1,2,…,n}

E.g. for n=3, A1, A2, A3 are mutually independent iff P(A1A2A3)=P(A1)P(A2)P(A3) and P(A1A2)=P(A1)P(A2) and P(A1A3)=P(A1)P(A3) and P(A2A3)=P(A2)P(A3)

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ji

iji

i APAP )()(

DISCRETE RANDOM VARIABLES

• If the set of all possible values of a r.v. X is a countable set, then X is called discrete r.v.

• The function f(x)=P(X=x) for x=x1,x2, … that assigns the probability to each value x is called probability density function (p.d.f.) or probability mass function (p.m.f.)

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Example

• Discrete Uniform distribution:

• Example: throw a fair die. P(X=1)=…=P(X=6)=1/6

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,...2,1N;N,...,2,1x;N

1)xX(P

CONTINUOUS RANDOM VARIABLES

• When sample space is uncountable (continuous)

• Example: Continuous Uniform(a,b)

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.bxaab

1)X(f

CUMULATIVE DENSITY FUNCTION (C.D.F.)

• CDF of a r.v. X is defined as F(x)=P(X≤x).

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JOINT DISCRETE DISTRIBUTIONS

• A function f(x1, x2,…, xk) is the joint pmf for some vector valued rv X=(X1, X2,…,Xk) iff the following properties are satisfied:

f(x1, x2,…, xk) 0 for all (x1, x2,…, xk)

and

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.1x,...,x,xf...

1x kxk21

MARGINAL DISCRETE DISTRIBUTIONS

• If the pair (X1,X2) of discrete random variables has the joint pmf f(x1,x2), then the marginal pmfs of X1 and X2 are

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12

21222111xx

xxfxf and xxfxf ,,

CONDITIONAL DISTRIBUTIONS

• If X1 and X2 are discrete or continuous random variables with joint pdf f(x1,x2), then the conditional pdf of X2 given X1=x1 is defined by

• For independent rvs,

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elsewhere. 0 f that such ,0xx,xf

x,xfxxf 11

1

2112

.

.

121

212

xfxxf

xfxxf

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EXPECTED VALUESLet X be a rv with pdf fX(x) and g(X) be a

function of X. Then, the expected value (or the mean or the mathematical expectation) of g(X)

Xx

X

g x f x , if X is discrete

E g Xg x f x dx, if X is continuous

providing the sum or the integral exists, i.e.,<E[g(X)]<.

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EXPECTED VALUES

• E[g(X)] is finite if E[| g(X) |] is finite.

Xx

X

g x f x < , if X is discrete

E g Xg x f x dx< , if X is continuous

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Laws of Expected Value E(c) = c E(X + c) = E(X) + c E(cX) = cE(X)

Laws of Variance V(c) = 0 V(X + c) = V(X) V(cX) = c2V(X)

Laws of Expected Value and Variance

Let X be a rv and c be a constant.

EXPECTED VALUE

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.

k

iii

k

iii XEaXaE

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If X and Y are independent,

YhEXgEYhXgE

The covariance of X and Y is defined as

)Y(E)X(E)XY(E

YEYXEXEY,XCov

EXPECTED VALUE

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If X and Y are independent,

0YXCov ,

The reverse is usually not correct! It is only correct under normal distribution.

If (X,Y)~Normal, then X and Y are independent iff

Cov(X,Y)=0

EXPECTED VALUE

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212121 2 XXCovXVarXVarXXVar ,

If X1 and X2 are independent,

2121 XVarXVarXXVar

CONDITIONAL EXPECTATION AND VARIANCE

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.continuous are Y and X if , dyxyyf

discrete. are Y and X if , xyyf

xYEy

22 xYExYExYVar

CONDITIONAL EXPECTATION AND VARIANCE

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YEXYEE

))X|Y(E(Var))X|Y(Var(E)Y(Var XX

(EVVE rule)

Proofs available in Casella & Berger (1990), pgs. 154 & 158

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SOME MATHEMATICAL EXPECTATIONS

• Population Mean: = E(X)

• Population Variance:

2 22 2 0Var X E X E X

(measure of the deviation from the population mean)

• Population Standard Deviation: 2 0

• Moments:* kk E X the k-th moment

k

k E X the k-th central moment

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This measure reflects the dispersion of all the observations

The variance of a population of size N x1, x2,…,xN

whose mean is is defined as

The variance of a sample of n observationsx1, x2, …,xn whose mean is is defined asx

N

)x( 2i

N1i2

N

)x( 2i

N1i2

1n

)xx(s

2i

n1i2

1n

)xx(s

2i

n1i2

The Variance

n

xx

ns i

ni

i

n

i

212

1

2 )(

1

1

MOMENT GENERATING FUNCTION

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xall

tx

xall

tx

tXX

discreteisXif)x(fe

.contisXifdx)x(fe

)e(E)t(M

The m.g.f. of random variable X is defined as

for t Є (-h,h) for some h>0.

Properties of m.g.f.

• M(0)=E[1]=1

• If a r.v. X has m.g.f. M(t), then Y=aX+b has a m.g.f.

•

• M.g.f does not always exists (e.g. Cauchy distribution)

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)at(Mebt

.derivativektheisMwhere)0(M)X(E th)k()k(k

CHARACTERISTIC FUNCTION

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xall

itx

xall

itx

itXX

discreteisXifxfe

contisXifdxxfe

eEt)(

.)(

)()(

The c.h.f. of random variable X is defined as

for all real numbers t. 1,12 ii

C.h.f. always exists.

Uniqueness

Theorem:

1.If two r.v.s have mg.f.s that exist and are equal, then they have the same distribution.

2.If two r.v.s have the same distribution, then they have the same m.g.f. (if they exist)

Similar statements are true for c.h.f.

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SOME DISCRETE PROBABILITY DISTRIBUTIONS

• Please review: Degenerate, Uniform, Bernoulli, Binomial, Poisson, Negative Binomial, Geometric, Hypergeometric, Extended Hypergeometric, Multinomial

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SOME CONTINUOUS PROBABILITY DISTRIBUTIONS

• Please review: Uniform, Normal (Gaussian), Exponential, Gamma, Chi-Square, Beta, Weibull, Cauchy, Log-Normal, t, F Distributions

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TRANSFORMATION OF RANDOM VARIABLES

• If X is an rv with pdf f(x), then Y=g(X) is also an rv. What is the pdf of Y?

• If X is a discrete rv, replace Y=g(X) whereever you see X in the pdf of f(x) by using the relation .

• If X is a continuous rv, then do the same thing, but now multiply with Jacobian.

• If it is not 1-to-1 transformation, divide the region into sub-regions for which we have 1-to-1 transformation.

)y(gx 1

CDF method

• Example: Let

Consider . What is the p.d.f. of Y?

• Solution:

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0xfore1)x(F x2

XeY

1yfory2)y(Fdy

d)y(f

1yfory1)y(lnF

)ylnX(P)ye(P)yY(P)y(F

3YY

2X

XY

M.G.F. Method

• If X1,X2,…,Xn are independent random variables with MGFs Mxi (t), then the MGF of is

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n

1iiXY )t(M)...t(M)t(M nX1XY

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THE PROBABILITY INTEGRAL TRANSFORMATION

• Let X have continuous cdf FX(x) and define the rv Y as Y=FX(x). Then,

Y ~ Uniform(0,1), that is,

P(Y y) = y, 0<y<1.

• This is very commonly used, especially in random number generation procedures.

SAMPLING DISTRIBUTION

• A statistic is also a random variable. Its distribution depends on the distribution of the random sample and the form of the function Y=T(X1, X2,…,Xn). The probability distribution of a statistic Y is called the sampling distribution of Y.

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SAMPLING FROM THE NORMAL DISTRIBUTION

Properties of the Sample Mean and Sample Variance

• Let X1, X2,…,Xn be a r.s. of size n from a N(,2) distribution. Then,

2) and are independent rvs.a X S

2) ~ , /b X N n

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12

1) ~ n

n Sc

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SAMPLING FROM THE NORMAL DISTRIBUTION

If population variance is unknown, we use sample variance:

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SAMPLING FROM THE NORMAL DISTRIBUTION

• The F distribution allows us to compare the variances by giving the distribution of

2 2 2 2

1, 12 2 2 2

/ /~

/ /X Y X X

n m

X Y Y Y

S S SF

S

• If X~Fp,q, then 1/X~Fq,p.

• If X~tq, then X2~F1,q.

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CENTRAL LIMIT THEOREMIf a random sample is drawn from any population, the

sampling distribution of the sample mean is approximately normal for a sufficiently large sample size. The larger the sample size, the more closely the sampling distribution of will resemble a normal distribution.

Random Sample

(X1, X2, X3, …,Xn)

Sample Mean Distribution

XX

Random Variable (Population) Distribution

as n

X

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Sampling Distribution of the Sample Mean

If X is normal, is normal.

If X is non-normal, is approximately normally distributed for sample size greater than or equal to 30.

X

2

2 or X Xn n

X

X

2 XX ~ N( , / n ) Z ~ N(0,1)

/ n