Stat 552 probability and statistics ii
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STAT 552 PROBABILITY AND STATISTICS II. INTRODUCTION Short review of S551. WHAT IS STATISTICS?.

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STAT 552 PROBABILITY AND STATISTICS II

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Stat 552 probability and statistics ii

STAT 552PROBABILITY AND STATISTICS II

INTRODUCTION

Short review of S551


What is statistics

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.


Basic definitions

BASIC DEFINITIONS

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

  • 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: A descriptive measure of a sample

  • VARIABLE: A characteristic of interest about each

    element of a population or sample.


Statistic

STATISTIC

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

    • are statistics.

    • are NOT.

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


Sample space

Sample Space

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

  • Determining the outcomes.

    • Build an exhaustive list of all possible outcomes.

    • Make sure the listed outcomes are mutually exclusive.


Random variables

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.


Descriptive statistics

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 examples

Types of data – examples


Stat 552 probability and statistics ii

PROBABILITY

POPULATION

SAMPLE

STATISTICAL

INFERENCE


Stat 552 probability and statistics ii

  • 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

Probability

P : S  [0,1]

Probability domain range

function


The calculus of probabilities

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)


Stat 552 probability and statistics ii

ODDS

  • The odds of an event A is defined by

  • It tells us how much more likely to see the

    occurrence of event A.


Odds ratio

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.


Conditional probability

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)


Conditional probability1

CONDITIONAL PROBABILITY


Bayes theorem

BAYES THEOREM

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


Independence

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)


Discrete random variables

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.)


Example

Example

  • Discrete Uniform distribution:

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


Continuous random variables

CONTINUOUS RANDOM VARIABLES

  • When sample space is uncountable (continuous)

  • Example: Continuous Uniform(a,b)


Cumulative density function c d f

CUMULATIVE DENSITY FUNCTION (C.D.F.)

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


Joint discrete distributions

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


Marginal discrete distributions

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


Conditional distributions

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,


Expected values

EXPECTED VALUES

Let 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)

providing the sum or the integral exists, i.e.,

<E[g(X)]<.


Expected values1

EXPECTED VALUES

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


Laws of expected value and variance

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

EXPECTED VALUE

If X and Y are independent,

The covariance of X and Y is defined as


Expected value1

EXPECTED VALUE

If X and Y are independent,

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 value2

EXPECTED VALUE

If X1 and X2 are independent,


Conditional expectation and variance

CONDITIONAL EXPECTATION AND VARIANCE


Conditional expectation and variance1

CONDITIONAL EXPECTATION AND VARIANCE

(EVVE rule)

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


Some mathematical expectations

SOME MATHEMATICAL EXPECTATIONS

  • Population Mean:  = E(X)

  • Population Variance:

(measure of the deviation from the population mean)

  • Population Standard Deviation:

  • Moments:


The variance

  • This measure reflects the dispersion of all the observations

  • The variance of a population of size N x1, x2,…,xN whose mean is m is defined as

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

The Variance


Moment generating function

MOMENT GENERATING FUNCTION

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

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)


Characteristic function

CHARACTERISTIC FUNCTION

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

for all real numbers t.

C.h.f. always exists.


Uniqueness

Uniqueness

Theorem:

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

  • 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.


Some discrete probability distributions

SOME DISCRETE PROBABILITY DISTRIBUTIONS

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


Some continuous probability distributions

SOME CONTINUOUS PROBABILITY DISTRIBUTIONS

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


Transformation of random variables

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.


Cdf method

CDF method

  • Example: Let

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

  • Solution:


M g f method

M.G.F. Method

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


The probability integral transformation

THE PROBABILITY INTEGRAL TRANSFORMATION

  • Let X have continuous cdfFX(x) and define the rvY 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

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.


Sampling from the normal distribution

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,


Sampling from the normal distribution1

SAMPLING FROM THE NORMAL DISTRIBUTION

If population variance is unknown, we use sample variance:


Sampling from the normal distribution2

SAMPLING FROM THE NORMAL DISTRIBUTION

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

  • If X~Fp,q, then 1/X~Fq,p.

  • If X~tq, then X2~F1,q.


Central limit theorem

X

Random Variable (Population) Distribution

Sample Mean Distribution

CENTRAL LIMIT THEOREM

If 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)


Sampling distribution of the sample mean

Sampling Distribution of the Sample Mean

If X is normal, is normal.

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


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