Stats 241 3 Probability Theory Summary Probability Axioms

Stats 241. 3 Probability Theory Summary

Probability

Axioms of Probability A probability measure P is defined on S by defining for each event E, P[E] with the following properties 1. 2. 3. P[E] ≥ 0, for each E. P[S] = 1.

Finite uniform probability space Many examples fall into this category 1. Finite number of outcomes 2. All outcomes are equally likely 3. To handle problems in case we have to be able to count. Count n(E) and n(S).

Techniques for counting

Basic Rule of counting Suppose we carry out k operations in sequence Let n 1 = the number of ways the first operation can be performed ni = the number of ways the ith operation can be performed once the first (i - 1) operations have been completed. i = 2, 3, … , k Then N = n 1 n 2 … nk = the number of ways the k operations can be performed in sequence.

Basic Counting Formulae 1. Permutations: How many ways can you order n objects n! 2. Permutations of size k (< n): How many ways can you choose k objects from n objects in a specific order

3. Combinations of size k ( ≤ n): A combination of size k chosen from n objects is a subset of size k where the order of selection is irrelevant. How many ways can you choose a combination of size k objects from n objects (order of selection is irrelevant)

Important Notes 1. In combinations ordering is irrelevant. Different orderings result in the same combination. 2. In permutations order is relevant. Different orderings result in the different permutations.

Rules of Probability

The additive rule P[A B] = P[A] + P[B] – P[A B] and P[A B] = P[A] + P[B] if P[A B] = f

The additive rule for more than two events and if Ai Aj = f for all i ≠ j. then

The Rule for complements for any event E

Conditional Probability, Independence and The Multiplicative Rule

The conditional probability of A given B is defined to be:

The multiplicative rule of probability and if A and B are independent. This is the definition of independence

The multiplicative rule for more than two events

Independence for more than 2 events

Definition: The set of k events A 1, A 2, … , Ak are called mutually independent if: P[Ai 1 ∩ Ai 2 ∩… ∩ Aim] = P[Ai 1] P[Ai 2] …P[Aim] For every subset {i 1, i 2, … , im } of {1, 2, …, k } i. e. for k = 3 A 1, A 2, … , Ak are mutually independent if: P[A 1 ∩ A 2] = P[A 1] P[A 2], P[A 1 ∩ A 3] = P[A 1] P[A 3], P[A 2 ∩ A 3] = P[A 2] P[A 3], P[A 1 ∩ A 2 ∩ A 3] = P[A 1] P[A 2] P[A 3]

Definition: The set of k events A 1, A 2, … , Ak are called pairwise independent if: P[Ai ∩ Aj] = P[Ai] P[Aj] for all i and j. i. e. for k = 3 A 1, A 2, … , Ak are pairwise independent if: P[A 1 ∩ A 2] = P[A 1] P[A 2], P[A 1 ∩ A 3] = P[A 1] P[A 3], P[A 2 ∩ A 3] = P[A 2] P[A 3], It is not necessarily true that P[A 1 ∩ A 2 ∩ A 3] = P[A 1] P[A 2] P[A 3]

Bayes Rule for probability

An generalization of Bayes Rule Let A 1, A 2 , … , Ak denote a set of events such that for all i and j. Then

Random Variables an important concept in probability

A random variable , X, is a numerical quantity whose value is determined be a random experiment

Definition – The probability function, p(x), of a random variable, X. For any random variable, X, and any real number, x, we define where {X = x} = the set of all outcomes (event) with X = x. For continuous random variables p(x) = 0 for all values of x.

Definition – The cumulative distribution function, F(x), of a random variable, X. For any random variable, X, and any real number, x, we define where {X ≤ x} = the set of all outcomes (event) with X ≤ x.

Discrete Random Variables For a discrete random variable X the probability distribution is described by the probability function p(x), which has the following properties

Graph: Discrete Random Variable p(x) a b

Continuous random variables For a continuous random variable X the probability distribution is described by the probability density function f(x), which has the following properties : 1. f(x) ≥ 0

Graph: Continuous Random Variable probability density function, f(x)

The distribution function F(x) This is defined for any random variable, X. F(x) = P[X ≤ x] Properties 1. 2. 3. F(-∞) = 0 and F(∞) = 1. F(x) is non-decreasing (i. e. if x 1 < x 2 then F(x 1) ≤ F(x 2) ) F(b) – F(a) = P[a < X ≤ b].

4. p(x) = P[X = x] =F(x) – F(x-) Here 5. If p(x) = 0 for all x (i. e. X is continuous) then F(x) is continuous.

6. For Discrete Random Variables F(x) is a non-decreasing step function with F(x) p(x)

7. For Continuous Random Variables F(x) is a non-decreasing continuous function with f(x) slope F(x) x To find the probability density function, f(x), one first finds F(x) then

Some Important Discrete distributions

The Bernoulli distribution

Suppose that we have a experiment that has two outcomes 1. Success (S) 2. Failure (F) These terms are used in reliability testing. Suppose that p is the probability of success (S) and q = 1 – p is the probability of failure (F) This experiment is sometimes called a Bernoulli Trial Let Then

The probability distribution with probability function is called the Bernoulli distribution p q = 1 - p

The Binomial distribution

We observe a Bernoulli trial (S, F) n times. Let X denote the number of successes in the n trials. Then X has a binomial distribution, i. e. where 1. p = the probability of success (S), and 2. q = 1 – p = the probability of failure (F)

The Poisson distribution • Suppose events are occurring randomly and uniformly in time. • Let X be the number of events occuring in a fixed period of time. Then X will have a Poisson distribution with parameter l.

The Geometric distribution Suppose a Bernoulli trial (S, F) is repeated until a success occurs. X = the trial on which the first success (S) occurs. The probability function of X is: p(x) =P[X = x] = (1 – p)x – 1 p = p qx - 1

The Negative Binomial distribution Suppose a Bernoulli trial (S, F) is repeated until k successes occur. Let X = the trial on which the kth success (S) occurs. The probability function of X is:

The Hypergeometric distribution Suppose we have a population containing N objects. Suppose the elements of the population are partitioned into two groups. Let a = the number of elements in group A and let b = the number of elements in the other group (group B). Note N = a + b. Now suppose that n elements are selected from the population at random. Let X denote the elements from group A. The probability distribution of X is

Continuous Distributions

Continuous random variables For a continuous random variable X the probability distribution is described by the probability density function f(x), which has the following properties : 1. f(x) ≥ 0

Graph: Continuous Random Variable probability density function, f(x)

Continuous Distributions The Uniform distribution from a to b

The Normal distribution (mean m, standard deviation s) s m

The Exponential distribution

The Weibull distribution A model for the lifetime of objects that do age.

The Weibull distribution with parameters a and b.

The Weibull density, f(x) (a = 0. 9, b = 2) (a = 0. 7, b = 2) (a = 0. 5, b = 2)

The Gamma distribution An important family of distributions

The Gamma distribution Let the continuous random variable X have density function: Then X is said to have a Gamma distribution with parameters a and l.

Graph: The gamma distribution (a = 2, l = 0. 9) (a = 2, l = 0. 6) (a = 3, l = 0. 6)

Comments 1. The set of gamma distributions is a family of distributions (parameterized by a and l). 2. Contained within this family are other distributions a. The Exponential distribution – in this case a = 1, the gamma distribution becomes the exponential distribution with parameter l. The exponential distribution arises if we are measuring the lifetime, X, of an object that does not age. It is also used a distribution for waiting times between events occurring uniformly in time. b. The Chi-square distribution – in the case a = n/2 and l = ½, the gamma distribution becomes the chi- square (c 2) distribution with n degrees of freedom. Later we will see that sum of squares of independent standard normal variates have a chi-square distribution, degrees of freedom = the number of independent terms in the sum of squares.

Expectation

Let X denote a discrete random variable with probability function p(x) (probability density function f(x) if X is continuous) then the expected value of X, E(X) is defined to be: and if X is continuous with probability density function f(x)

Expectation of functions Let X denote a discrete random variable with probability function p(x) then the expected value of X, E[g (X)] is defined to be: and if X is continuous with probability density function f(x)

Moments of a Random Variable

the kth moment of X : • The first moment of X , m = m 1 = E(X) is the center of gravity of the distribution of X. • The higher moments give different information regarding the distribution of X.

the kth central moment of X

Moment generating functions

Definition Let X denote a random variable, Then the moment generating function of X , m. X(t) is defined by:

Properties 1. m. X(0) = 1 2. 3.

4. Let X be a random variable with moment generating function m. X(t). Let Y = b. X + a Then m. Y(t) = mb. X + a(t) = E(e [b. X + a]t) = eat. E(e X[ bt ]) = eatm. X (bt) 5. Let X and Y be two independent random variables with moment generating function m. X(t) and m. Y(t). Then m. X+Y(t) = E(e [X + Y]t) = E(e Xt e Yt) = E(e Xt) E(e Yt) = m. X (t) m. Y (t)

6. Let X and Y be two random variables with moment generating function m. X(t) and m. Y(t) and two distribution functions FX(x) and FY(y) respectively. Let m. X (t) = m. Y (t) then FX(x) = FY(x). This ensures that the distribution of a random variable can be identified by its moment generating function

M. G. F. ’s - Continuous distributions

M. G. F. ’s - Discrete distributions

Note: The distribution of a random variable X can be described by:

Jointly distributed Random variables Multivariate distributions

Discrete Random Variables

The joint probability function; p(x, y) = P[X = x, Y = y]

Continuous Random Variables

Definition: Two random variable are said to have joint probability density function f(x, y) if

Marginal and conditional distributions

Marginal Distributions (Discrete case): Let X and Y denote two random variables with joint probability function p(x, y) then the marginal density of X is the marginal density of Y is

Marginal Distributions (Continuous case): Let X and Y denote two random variables with joint probability density function f(x, y) then the marginal density of X is the marginal density of Y is

Conditional Distributions (Discrete Case): Let X and Y denote two random variables with joint probability function p(x, y) and marginal probability functions p. X(x), p. Y(y) then the conditional density of Y given X = x conditional density of X given Y = y

Conditional Distributions (Continuous Case): Let X and Y denote two random variables with joint probability density function f(x, y) and marginal densities f. X(x), f. Y(y) then the conditional density of Y given X = x conditional density of X given Y = y

The bivariate Normal distribution

Let where This distribution is called the bivariate Normal distribution. The parameters are m 1, m 2 , s 1, s 2 and r.

Surface Plots of the bivariate Normal distribution

Marginal distributions 1. The marginal distribution of x 1 is Normal with mean m 1 and standard deviation s 1. 2. The marginal distribution of x 2 is Normal with mean m 2 and standard deviation s 2.

Conditional distributions 1. The conditional distribution of x 1 given x 2 is Normal with: mean and standard deviation 2. The conditional distribution of x 2 given x 1 is Normal with: mean standard deviation and

Independence

Definition: Two random variables X and Y are defined to be independent if if X and Y are discrete if X and Y are continuous

multivariate distributions k≥ 2

Definition Let X 1, X 2, …, Xn denote n discrete random variables, then p(x 1, x 2, …, xn ) is joint probability function of X 1, X 2, …, Xn if

Definition Let X 1, X 2, …, Xk denote k continuous random variables, then f(x 1, x 2, …, xk ) is joint density function of X 1, X 2, …, Xk if

The Multinomial distribution Suppose that we observe an experiment that has k possible outcomes {O 1, O 2, …, Ok } independently n times. Let p 1, p 2, …, pk denote probabilities of O 1, O 2, …, Ok respectively. Let Xi denote the number of times that outcome Oi occurs in the n repetitions of the experiment.

The joint probability function of: is called the Multinomial distribution

The Multivariate Normal distribution Recall the univariate normal distribution the bivariate normal distribution

The k-variate Normal distribution where

Marginal distributions

Definition Let X 1, X 2, …, Xq+1 …, Xk denote k discrete random variables with joint probability function p(x 1, x 2, …, xq+1 …, xk ) then the marginal joint probability function of X 1, X 2, …, Xq is

Definition Let X 1, X 2, …, Xq+1 …, Xk denote k continuous random variables with joint probability density function f(x 1, x 2, …, xq+1 …, xk ) then the marginal joint probability function of X 1, X 2, …, Xq is

Conditional distributions

Definition Let X 1, X 2, …, Xq+1 …, Xk denote k discrete random variables with joint probability function p(x 1, x 2, …, xq+1 …, xk ) then the conditional joint probability function of X 1, X 2, …, Xq given Xq+1 = xq+1 , …, Xk = xk is

Definition Let X 1, X 2, …, Xq+1 …, Xk denote k continuous random variables with joint probability density function f(x 1, x 2, …, xq+1 …, xk ) then the conditional joint probability function of X 1, X 2, …, Xq given Xq+1 = xq+1 , …, Xk = xk is

Definition – Independence of sets of vectors Let X 1, X 2, …, Xq+1 …, Xk denote k continuous random variables with joint probability density function f(x 1, x 2, …, xq+1 …, xk ) then the variables X 1, X 2, …, Xq are independent of Xq+1, …, Xk if A similar definition for discrete random variables.

Definition – Mutual Independence Let X 1, X 2, …, Xk denote k continuous random variables with joint probability density function f(x 1, x 2, …, xk ) then the variables X 1, X 2, …, Xk are called mutually independent if A similar definition for discrete random variables.

Expectation for multivariate distributions

Definition Let X 1, X 2, …, Xn denote n jointly distributed random variable with joint density function f(x 1, x 2, …, xn ) then

Some Rules for Expectation

Thus you can calculate E[Xi] either from the joint distribution of X 1, … , Xn or the marginal distribution of Xi. The Linearity property

3. (The Multiplicative property) Suppose X 1, … , Xq are independent of Xq+1, … , Xk then In the simple case when k = 2 if X and Y are independent

Some Rules for Variance

Tchebychev’s inequality Ex:

Note: If X and Y are independent, then

The correlation coefficient r. XY 2. if there exists a and b such that where r. XY = +1 if b > 0 and r. XY = -1 if b< 0

Some other properties of variance

4. Variance: Multiplicative Rule for independent random variables Suppose that X and Y are independent random variables, then:

Mean and Variance of averages Let X 1, … , Xn be n mutually independent random variables each having mean m and standard deviation s (variance s 2). Let Then and

The Law of Large Numbers Let X 1, … , Xn be n mutually independent random variables each having mean m. Let Then for any d > 0 (no matter how small)

Conditional Expectation:

Definition Let X 1, X 2, …, Xq+1 …, Xk denote k continuous random variables with joint probability density function f(x 1, x 2, …, xq+1 …, xk ) then the conditional joint probability function of X 1, X 2, …, Xq given Xq+1 = xq+1 , …, Xk = xk is

Definition Let U = h( X 1, X 2, …, Xq+1 …, Xk ) then the Conditional Expectation of U given Xq+1 = xq+1 , …, Xk = xk is Note this will be a function of xq+1 , …, xk.

A very useful rule Let (x 1, x 2, … , xq, y 1, y 2, … , ym) = (x, y) denote q + m random variables. Then

Functions of Random Variables

Methods for determining the distribution of functions of Random Variables 1. Distribution function method 2. Moment generating function method 3. Transformation method

Distribution function method Let X, Y, Z …. have joint density f(x, y, z, …) Let W = h( X, Y, Z, …) First step Find the distribution function of W G(w) = P[W ≤ w] = P[h( X, Y, Z, …) ≤ w] Second step Find the density function of W g(w) = G'(w).

Use of moment generating functions 1. Using the moment generating functions of X, Y, Z, …determine the moment generating function of W = h(X, Y, Z, …). 2. Identify the distribution of W from its moment generating function This procedure works well for sums, linear combinations, averages etc.

Let x 1, x 2, … denote a sequence of independent random variables Sums Let S = x 1 + x 2 + … + xn then Linear Combinations Let L = a 1 x 1 + a 2 x 2 + … + anxn then

Arithmetic Means Let x 1, x 2, … denote a sequence of independent random variables coming from a distribution with moment generating function m(t)

The Transformation Method Theorem Let X denote a random variable with probability density function f(x) and U = h(X). Assume that h(x) is either strictly increasing (or decreasing) then the probability density of U is:

The Transfomation Method Theorem (many variables) Let x 1, x 2, …, xn denote random variables with joint probability density function f(x 1, x 2, …, xn ) Let u 1 = h 1(x 1, x 2, …, xn). u 2 = h 2(x 1, x 2, …, xn). un = hn(x 1, x 2, …, xn). define an invertible transformation from the x’s to the u’s

Then the joint probability density function of u 1, u 2, …, un is given by: where Jacobian of the transformation

Some important results Distribution of functions of random variables

The method used to derive these results will be indicated by: 1. DF - Distribution Function Method. 2. MGF - Moment generating function method 3. TF - Transformation method

Student’s t distribution Let Z and U be two independent random variables with: 1. Z having a Standard Normal distribution and 2. U having a c 2 distribution with n degrees of freedom then the distribution of: is: DF

The Chi-square distribution Let Z 1, Z 2, … , Zv be v independent random variables having a Standard Normal distribution, then has a c 2 distribution with n degrees of freedom. DF MGF for n = 1 for n > 1

Distribution of the sample mean Let x 1, x 2, …, xn denote a sample from the normal distribution with mean m and variance s 2. then has a Normal distribution with: MGF

The Central Limit theorem If x 1, x 2, …, xn is a sample from a distribution with mean m, and standard deviations s, then if n is large has a normal distribution with mean and variance MGF

Distribution of sums of Gamma R. V. ’s Let X 1, X 2, … , Xn denote n independent random variables each having a gamma distribution with parameters (l, ai), i = 1, 2, …, n. Then W = X 1 + X 2 + … + Xn has a gamma distribution with parameters (l, a 1 + a 2 +… + an). MGF Distribution of a multiple of a Gamma R. V. Suppose that X is a random variable having a gamma distribution with parameters (l, a). Then W = a. X has a gamma distribution with parameters (l/a, a). MGF

Distribution of sums of Binomial R. V. ’s Let X 1, X 2, … , Xk denote k independent random variables each having a binomial distribution with parameters (p, ni), i = 1, 2, …, k. Then W = X 1 + X 2 + … + Xk has a binomial distribution with parameters (p, n 1 + n 2 +… + nk). MGF Distribution of sums of Negative Binomial R. V. ’s Let X 1, X 2, … , Xn denote n independent random variables each having a negative binomial distribution with parameters (p, ki), i = 1, 2, …, n. Then W = X 1 + X 2 + … + Xn has a negative binomial distribution with parameters (p, k 1 + k 2 +… + kn). MGF

Beyond Stats 241 Courses that can be taken after Stats 241

Statistics

What is Statistics? It is the major mathematical tool of scientific inference – methods for drawing conclusion from data. Data that is to some extent corrupted by some component of random variation (random noise)

In both Statistics and Probability theory we are concerned with studying random phenomena

In probability theory The model is known and we are interested in predicting the outcomes and observations of the phenomena. model outcomes and observations

In statistics The model is unknown the outcomes and observations of the phenomena have been observed. We are interested in determining the model from the observations outcomes and observations model

Example - Probability A coin is tossed n = 100 times We are interested in the observation, X, the number of times the coin is a head. Assuming the coin is balanced (i. e. p = the probability of a head = ½. )

Example - Statistics We are interested in the success rate, p, of a new surgical procedure. The procedure is performed n = 100 times. X, the number of successful times the procedure is performed is 82. The success rate p is unknown.

If the success rate p was known. Then This equation allows us to predict the value of the observation, X.

In the case when the success rate p was unknown. Then the following equation is still true the success rate We will want to use the value of the observation, X = 82 to make a decision regarding the value of p.

Introductory Statistics Courses Non calculus Based Stats 244. 3 Stats 245. 3 Calculus Based Stats 242. 3

Stats 244. 3 Statistical concepts and techniques including graphing of distributions, measures of location and variability, measures of association, regression, probability, confidence intervals, hypothesis testing. Students should consult with their department before enrolling in this course to determine the status of this course in their program. Prerequisite(s): A course in a social science or Mathematics A 30.

Stats 245. 3 An introduction to basic statistical methods including frequency distributions, elementary probability, confidence intervals and tests of significance, analysis of variance, regression and correlation, contingency tables, goodness of fit. Prerequisite(s): MATH 100, 101, 102, 110 or STAT 103.

Stats 242. 3 Sampling theory, estimation, confidence intervals, testing hypotheses, goodness of fit, analysis of variance, regression and correlation. Prerequisite(s): MATH 110, 116 and STAT 241.

Stats 244 and 245 • do not require a calculus prerequisite • are Recipe courses Stats 242 • does require calculus and probability (Stats 241) as a prerequisite • More theoretical class – You learn techniques for developing statistical procedures and thoroughly investigating the properties of these procedures

Statistics Courses beyond Stats 242. 3

STAT 341. 3 Probability and Stochastic Processes 1/2(3 L-1 P) Prerequisite(s): STAT 241. Random variables and their distributions; independence; moments and moment generating functions; conditional probability; Markov chains; stationary time-series.

STAT 342. 3 Mathematical Statistics 1(3 L-1 P) Prerequisite(s): MATH 225 or 276; STAT 241 and 242. Probability spaces; conditional probability and independence; discrete and continuous random variables; standard probability models; expectations; moment generating functions; sums and functions of random variables; sampling distributions; asymptotic distributions. Deals with basic probability concepts at a moderately rigorous level. Note: Students with credit for STAT 340 may not take this course for credit.

STAT 344. 3 Applied Regression Analysis 1/2(3 L-1 P) Prerequisite(s): STAT 242 or 245 or 246 or a comparable course in statistics. Applied regression analysis involving the extensive use of computer software. Includes: linear regression; multiple regression; stepwise methods; residual analysis; robustness considerations; multicollinearity; biased procedures; nonlinear regression. Note: Students with credit for ECON 404 may not take this course for credit. Students with credit for STAT 344 will receive only half credit for ECON 404.

STAT 345. 3 Design and Analysis of Experiments 1/2(3 L-1 P) Prerequisite(s): STAT 242 or 245 or 246 or a comparable course in statistics. An introduction to the principles of experimental design and analysis of variance. Includes: randomization, blocking, factorial experiments, confounding, random effects, analysis of covariance. Emphasis will be on fundamental principles and data analysis techniques rather than on mathematical theory.

STAT 346. 3 Multivariate Analysis 1/2(3 L-1 P) Prerequisite(s): MATH 266, STAT 241, and 344 or 345. The multivariate normal distribution, multivariate analysis of variance, discriminant analysis, classification procedures, multiple covariance analysis, factor analysis, computer applications.

STAT 347. 3 Non Parametric Methods 1/2(3 L-1 P) Prerequisite(s): STAT 242 or 245 or 246 or a comparable course in statistics. An introduction to the ideas and techniques of non-parametric analysis. Includes: one, two and K samples problems, goodness of fit tests, randomness tests, and correlation and regression.

STAT 348. 3 Sampling Techniques 1/2(3 L-1 P) Prerequisite(s): STAT 242 or 245 or 246 or a comparable course in statistics. Theory and applications of sampling from finite populations. Includes: simple random sampling, stratified random sampling, cluster sampling, systematic sampling, probability proportionate to size sampling, and the difference, ratio and regression methods of estimation.

STAT 349. 3 Time Series Analysis 1/2(3 L-1 P) Prerequisite(s): STAT 241, and 344 or 345. An introduction to statistical time series analysis. Includes: trend analysis, seasonal variation, stationary and non-stationary time series models, serial correlation, forecasting and regression analysis of time series data.

STAT 442. 3 Statistical Inference 2(3 L-1 P) Prerequisite(s): STAT 342. Parametric estimation, maximum likelihood estimators, unbiased estimators, UMVUE, confidence intervals and regions, tests of hypotheses, Neyman Pearson Lemma, generalized likelihood ratio tests, chi-square tests, Bayes estimators.

STAT 443. 3 Linear Statistical Models 2(3 L-1 P) Prerequisite(s): MATH 266, STAT 342, and 344 or 345. A rigorous examination of the general linear model using vector space theory. Includes: generalized inverses; orthogonal projections; quadratic forms; Gauss-Markov theorem and its generalizations; BLUE estimators; Non-full rank models; estimability considerations.