Let us denote by S the event that the weather is good in summer and by H the event that it is good during the harvesting time. By the completeness axiom, these probabilities sum to 1. As usual, this information can be displayed in a table. Therefore we have two more tables Table 4.

The probabilities from Table 4. Thus, row sums of the probabilities in the central part of Table 4. Similarly, column sums of the probabilities in the central part of Table 4. The customary name for own probabilities, marginal probabilities, reflects their placement in the table but not their role.

Summation signs. This is the place to learn the techniques of working with summation signs. In the short expression we indicate the index that changes, i , and that it runs from the lower limit of summation, 1, to the upper one, n.

If you have problems working with summation signs, replace them with extended expressions, do the algebra, and then convert everything back. The answer to Exercise 4. We are left with one line of Table 4. But we know that there are no missing events. To satisfy the completeness axiom, we can divide both sides of 4. The reason is that occurrence of S provides us with additional information.

This process of updating our beliefs as new information arrives is one of the fundamental ideas of the theory of probabilities. Conditional probabilities are defined axiomatically and the subsequent theory justifies their introduction. Multiplication rule. Try to see the probabilistic meaning of this equation. Denote M the event that you obtain a passing grade in Math and by S the event that you obtain a passing grade in Stats.

Suppose that Math is a prerequisite for Stats. The fact that you have not failed Math tells something about your abilities and increases the chance of passing Stats. Events A and B are called independent if occurrence of one of them does not influence in any way the chances of occurring of the other. For example, what happens to the coin has nothing to do with what happens to the die.

Formal definition. This definition is applied in two ways. Always write down what is given and what you need to get. When rearranging expressions indicate what rule you are applying, until everything becomes trivial.

Now try to move the strip A left and right and the strip B up and down. Independence takes a physical meaning: movements in mutually orthogonal directions do not affect each other. Before occurrence of the event A , an expert forms an opinion about likelihood of the event B. That opinion, embodied in P B , is called a prior probability. By the Bayes theorem, updating is accomplished P A B through multiplication of the prior probability by the factor.

P A A simple way to complicate the matters. Suppose B1 , Have a look at Exercises in NCT 4. In the end of each chapter, starting from this one, there is a unit called Questions for Repetition, where, in addition to theoretical questions, I indicate a minimum of exercises from NCT.

To write down the conditions, you need a good notation. Avoid abstract notation and use something that reminds you of what you are dealing with, like A for Anthropology class. When looking for a solution of a problem, write down all the relevant theoretical facts. Omit only what is easy; after a while everything will become easy. Exercises in the end of Sections 4. This is where writing down all the relevant equations is particularly useful.

Remember what you need to find. In general, to find n unknowns it is sufficient to have n equations involving them. One of my students was able to solve those exercises without any knowledge of conditional and joint probabilities, just using common sense and the percentage interpretation of probabilities.

Learn to save your time by concentrating on ideas. Everybody finds difficult exercises that require orderings and combinations there are many of them in the end of Section 4. Solve more of them, until you feel comfortable. What do we mean by set operations? Prove de Morgan laws. Prove and illustrate on a Venn diagram equation 4.

Prove the formulas for orderings and combinations 6. Using a simple example, in one block give all definitions and identities related to joint and marginal probabilities.

Solve Exercise 4. Derive the complex form of the Bayes theorem 4. The minimum required: Exercises 4. Repeat everything with proofs. Look for those minor or significant details you missed earlier. Make a list of facts in a logical order. This chapter will be more challenging than all the previous taken together, and you will need to be in a perfect shape to conquer it. Unit 5. A random variable is a variable whose values cannot be predicted. For example, we can associate random variables with the coin and die.

On the other hand, to know the value of sin x it suffices to plug the argument x in. Short formal definition. A random variable is a pair: values plus probabilities. Working definition of a discrete random variable. A discrete random variable is defined by Table 5. Thus, in the first column we have real numbers.

This is important to be able to use arithmetic operations. This is a simplification. In Unit 5. What is your expected loss? Sometimes I ask questions before giving formal definitions, to prompt you to guess. Guessing and inductive reasoning are important parts of our thinking process.

In more complex situations there may be several competing ideas, and some of them may not be clear, looking like unfamiliar creatures in the dark. Looking for them and nurturing them until they shape up so that you can test them is a great skill. This is where you need to collect all the relevant information, mull it over again and again, until it becomes your second nature. This process is more time- consuming than getting a ready answer from your professor or classmate but the rewards are also greater.

The general idea is that it is a weighted sum of the values of the variable of interest the loss function. We attach a higher importance to the value that has a higher probability. In research ideas go before definitions. Expected value of a discrete random variable. And it is not a product of E and X! Recall the procedure for finding relative frequencies.

Denote y1 , Equal values are joined in groups. Let x1 , Their total is, clearly, n. In this context, the sample mean Y , the grouped data formula in the third line and the expected value are the same thing.

The weighted mean formula from NCT see their equation 3. Note also that when data are grouped by intervals, as in Table 3. Therefore 5. Suppose that X , Y are two discrete random variables with the same probability distribution p1 , Let a , b be real numbers. The detailed definition is given by the next table. Table 5. The situation when the probability distributions are different is handled using joint probabilities see Unit 5. Operations with vectors are defined by the last three columns of Table 5.

These formal definitions plus the geometric interpretation considered next is all you need to know about vectors. We live in the space of three-dimensional vectors. All our intuition coming from day-to-day geometrical experience carries over to the n -dimensional case. The rule itself comes from physics: if two forces are applied to a point, their resultant force is found by the parallelogram rule. Scaling by a negative number means, additionally, reverting the direction of X.

Problem statement. This is one of those straightforward proofs when knowing the definitions and starting with the left-hand side is enough to arrive at the result. Using the definitions in Table 5. Chewing and digesting Going back. A portfolio contains n1 shares of stock 1 whose price is S1 and n2 shares of stock 2 whose price is S 2.

Stock prices fluctuate and are random variables. Numbers of shares are assumed fixed and are deterministic. What is the expected value of the portfolio? Going sideways. In the next two propositions the words are the same, just their order is different: i In the city of X, for every man M there is a woman W such that W is a girlfriend of M.

This property is called homogeneity of degree 1 you can pull the constant out of the expected value sign. This is called additivity. Visually, a random variable is a 1-D table with values and probabilities. A pair of random variables is represented by a 2-D table similar to Table 4. Since we want to work with variables whose probability distributions are not necessarily the same, a table of type Table 5.

We deal with two random variables separately represented by the following tables Table 5. Now we arrange a 2-D table putting a pair of values xi , y j and corresponding probability pij in the same cell: Table 5.

The variable X is realized in Table 5. What is its realization in terms of Table 5. Exercise 5. We say that X and Y are independent if what happens to X does not influence Y in any way. Consider a universal statement: all students in a given class are younger than Its opposite is: there are students in a given class who are at least 30 years old, which is an existence statement.

This is a general rule of logic: rejecting a universal statement you obtain an existence one, and vice versa. Thus, we say that X and Y are dependent if at least one of the equations 5. Multiplicativity of expected values. Earlier you were asked to rewrite the proof of linearity of means using summation signs.

All operators in math that are linear may have nonlinear properties only under special conditions. Independence is such a special condition. Similarly, the complete expression for the probabilities from Table 5. This formalism is necessary to understand some equations in NCT. Therefore we shall be using linearity of means very often. Therefore cov X , Y is the expected value of the product of these two deviations. Explain all formulas for yourself in this way.

This is true with some caveats. We keep covariance around mainly for its algebraic properties. Property 1. We start by writing out the left side of 5. Write out and prove this property. You can notice the importance of using parentheses and brackets. Property 2. Uncorrelatedness is close to independence, so the intuition is the same: one variable does not influence the other. You can also say that there is no statistical relationship between uncorrelated variables.

Property 3. When this is true, we also say that B is necessary for A. Visually, the set of all objects satisfying A is a subset of all objects satisfying B when A is sufficient for B. An interesting question is whether independence is equivalent to uncorrelatedness or, put it differently, if there exist uncorrelated pairs which are not independent.

The answer is Yes. This Y is not independent of X because knowledge of the latter implies knowledge of the former. Property 4. Correlation with a constant. Property 5. For nonlinear operations, of type V X , the argument must be put in parentheses.

Variance of a linear combination. Characterization of all variables with vanishing variance. We want to know exactly which variables have zero variance. Since all probabilities are positive, from 5.

This implies that all values are equal to the mean, which is a constant. Since variance is a nonlinear operator, it is additive only under special circumstances, when the variables are uncorrelated or, in particular, when they are independent. In general, there are two square roots of a positive number, one positive and the other negative. The positive one is called an arithmetic square root. Absolute values. By squaring both sides in 5.

This is the equation we need right now. Cauchy-Schwarz inequality. To exclude the trivial case, let X , Y be nonconstant. We see that f t is a parabola with branches looking upward because the senior coefficient V X is positive. Do you think this proof is tricky? During the long history of development of mathematics mathematicians have invented many tricks, small and large.

No matter how smart you are, you cannot reinvent all of them. By the way, the definition of what is tricky and what is not is strictly personal and time-dependent.

Suppose random variables X , Y are not constant. Interpretation of extreme cases. Further, we can establish the sign of the number a. Plugging this in 5. The proof of 2 is left as an exercise. Suppose we want to measure correlation between weight W and height H of people. The motivation here is purely mathematical. Digression on linear transformations. From Table 5. Correspondingly, the histogram moves right or left as a rigid body the distances between values do not change.

In case of stretching its variance increases and in case of contraction it decreases. The transformation of X defined in 5. A Bernoulli variable is the variable that describes an unfair coin. A Bernoulli variable is defined by the table: Table 5. Variance of the Bernoulli variable. Note that V B is a parabola with branches looking downward. Take any book in English.

All alphabetic characters in it are categorized as either consonants or vowels. Denote by p the percentage of consonants. We assign 1 to a consonant and 0 to a vowel. The experiment consists in randomly selecting an alphabetic character from the book.

Then the outcome is 1 with probability p and 0 with probability q. In this context, the population is the set of consonants and vowels, along with their percentages p and q. This description fits the definition from Section 1. Another definition from the same section says: a sample is an observed subset of population values with sample size given by n. It is important to imagine the sampling process step by step.

To obtain a sample in our example we can randomly select n characters and write the sample as X1 , We consider the most important two. Case I. We randomly select X1 from the whole book. Then we randomly select X 2 , again from the whole book. We continue like this following two rules: 1 each selection is random and 2 the selection is made from the whole book.

The second condition defines what is called a sampling with replacement: to restore the population, selected elements are returned to it. S n , being a sum of unities and zeros, will be some integer from the range [0, n ] and X will be some fraction from [0,1]. The subtlety of the theoretical argument in statistics is that X 1 , Let us rewind the movie and think about the selection process again.

The associated random variable is denoted X1 and, clearly, is described by the same table as B. Similarly, you think about what possibilities you would face in all other cases and come to the conclusion that all drawings are described by the same table, just the names of the variables will be different.

Random variables that have the same distributions that is, values plus probabilities are called identically distributed. In this case the population we draw the sample from is called a parent population. If all are zeros, S n is zero. If all are unities, S n will be equal to n. In general, S n may take any integer value from 0 to n , inclusive. Now the next definition must be clear.

A binomial variable is a sum of n independent identically distributed Bernoulli variables. Case II. Think about an opinion poll on a simple question requiring a simple answer: Yes or No. Suppose everybody has a definite opinion on the matter, and denote p the percentage of voters who have an affirmative answer.

A team is sent out to conduct a survey. They record 1 for Yes and 0 for No. The population will be again Bernoulli but there are problems with the sample. In essence, after having randomly selected the first individual, you cross out that individual from the population before selecting the next one. This implies two things: the observations are dependent and their distributions are not identical.

The general statistical principle is that removing one element from a large population does not change its characteristics very much. So in the case under consideration, the observations can be assumed approximately i. But there is a much easier way to do that, which should be obvious, if not now then at least after reading the proof below. It is as handy as it is indispensable in a number of applications.

In the context of drawing from a Bernoulli population S n is called a number of successes because only unities contribute to it. For variance we use also independence. Due to independence, all interaction terms vanish and variance is additive. The quantity on the left is called a standard error n of X.

This is a general statistical fact: increasing the sample is always good. Let us derive the distribution of S 3. The idea is to list all possible combinations of values of X 1 , X 2 , X 3 , find the corresponding values of S 3 and then calculate their probabilities. This is done in two steps. In mathematical exposition ideas sometimes are not stated. The presumption is that you get the idea after reading the proof.

In your mind, ideas should always go first. The next step is to collect equal values of S 3. This is done using the additivity rule. Thus the result is Table 5. Case of general n. We have to find out how many such sequences exist for the given x. This question can be rephrased as follows: in how many different ways x places where the unities will be put can be chosen out of n places?

Example 5. The utility company would like to know the likelihood of a jump in electricity consumption tomorrow. What happens to the probability?

Let X be a random variable and x a real number. As the event becomes wider, the probability increases. Such probability can be expressed in terms of the distribution function. However, in case of a discrete random variable the argument x is often restricted to the values taken by the variable.

The sums arising here are called cumulative probabilities. We summarize our findings in a table: Table 5. Therefore we introduce it axiomatically and then start looking for real-world applications. Taylor decomposition. Unlike marsupials, they are considered simple. The idea is to represent other functions as linear combinations of monomials.

Linear and quadratic functions are special cases of polynomials. If infinite linear combinations, called Taylor decompositions, are allowed, then the set of functions representable using monomials becomes much wider.

Hence, we can define a x! This variable is called a Poisson distribution. An application should be a variable which can potentially take nonnegative integer values, including very large ones. Normally, a mobile network operator, such as T-Mobile, has a very large number of subscribers. Let X denote the number of all customers of T-Mobile who use their mobile service at a given moment.

It is random, it takes nonnegative integer values and the number of customers using their mobile phones simultaneously can be very large. Finally, decisions by customers to use their cell phones can be considered independent, unless there is a disaster.

Since all the accompanying logic has been cut off, the result is a complete mystery. This is another indication as to where it can be applied. Before taking up the last point, we look at an example. In any given year the probability that any single policy will result in a claim is 0. Find the probability that at least three claims are made in a given year. On the exam you may not have enough time to calculate the answer. Show that you know the material by introducing the right notation and telling the main idea.

Ideas go first! Denote S n the binomial variable. You will die before you finish raising 0. The distinctive feature of 5. This small number is multiplied by combinations Cxn which, when n is very large, may be for a simple calculator the same thing as infinity. However, the product of the two numbers, one small and another large, may be manageable. Mathematicians are reasonably lazy. When they see such numbers as in 5. The device for the problem at hand is called a Poisson approximation to the binomial distribution and sounds as follows.

In fact, the portfolio analysis is a little bit different. To explain the difference, we start with fixing two points of view. If I want to sell it, I am interested in knowing its market value. In this situation the numbers of shares in my portfolio, which are constant, and the market prices of stocks, which are random, determine the market value of the portfolio, defined in Unit 5. The value of the portfolio is a linear combination of stock prices.

Being a gambler, I am not interested in holding a portfolio. I am thinking about buying a portfolio of stocks now and selling it, say, in a year at price M 1. M0 M 0 is considered deterministic current prices are certain and M 1 is random future prices are unpredictable. The rate of return thus is random. Main result. The rate of return on the portfolio is a linear combination of the rates of return on separate assets.

As it often happens in economics and finance, this result depends on how one understands the things. The initial amount M 0 is invested in n assets. Then using 5. M0 M0 M0 M0 This is the main result. Once you know this equation you can find the mean and variance of the rate of return on the portfolio in terms of shares and rates of return on assets.

List and prove all properties of means. List and prove all properties of covariances. List and prove all properties of variances. List and prove all properties of standard deviations.

List and prove all properties of the correlation coefficient including the statistical interpretation from Table 3. How and why do you standardize a variable? Define and derive the properties of the Bernoulli variable. Define a binomial distribution and derive its mean and variance with explanations.

Prove equation 5. Define a distribution function, describe its geometric behavior and prove the interval formula. Define the Poisson distribution and describe without proof how it is applied to the binomial distribution. Minimum required: Exercises 5. The definition of the distribution function given there is general and applies equally to discrete and continuous random variables. It has been mentioned that in case of a discrete variable X the argument x of its distribution function is usually restricted to the values of X.

Now we are going to look into the specifics of the continuous case. Let the function f be defined on the segment [ a , b ]. A2 a c b A1 Figure 6. Thus, for the function in b Figure 6. If we change any of a, b, f , the integral will change. The argument x does not affect the value of the integral, even though it is present in the notation of the integral b 6. The variable of integration t is included here to facilitate some manipulations with integrals.

Let f , g be two functions and let c, d be two numbers. This is one of those cases when you can use the cow-and-goat principle: here is a cow and there is a goat; they are quite similar and can be used, to a certain extent, for the same purposes. Regarding the probabilities, we have established two properties: they are percentages and they satisfy the completeness axiom.

We want an equivalent of these notions and properties in the continuous case. Let X be a continuous random variable and let FX be its distribution function.

All continuous random variables that do not satisfy this existence requirement are left out in this course. From 6. Use the interval formula 5. The probability of X taking any specific value is zero because by 6. If the density is negative at some point x0 , we can take a and b close to that point and then by the interval formula 6. Differences between the cow and goat.

In the continuous case b the values of the density pX t mean nothing and can be larger than 1. Exercise 6. For example, you can take: i distributions of income in a wealthy neighborhood and in a poor neighborhood, ii distributions of temperature in winter and summer in a given geographic location; iii distributions of electricity consumption in two different locations at the same time of the year.

In the preamble of the definition you will see the argument of type used in the times of Newton and Leibniz, the fathers of calculus. It is still used today in inductive arguments by modern mathematicians. The idea of the next construction is illustrated in Figure 6. The segment [ a , b ] is divided into many small segments, denoted [t0 , t1 ], Therefore 6. Let X be a continuous random variable that takes values in [ a , b ]. Let us divide [ a , b ] into equal parts 6.

If we have a sample of n observations on X , they can be grouped in batches depending on which subsegment they fall into. By the interval formula 6. All properties of means, covariance and variance derived from that definition are absolutely the same as in the discrete case. Also, for 10 we need the definition of a function of a random variable, and this is better explained following the cow-and-goat principle.

Table 6. Example 6. Do you think this probability at some points is higher than at others? Fix some segment [ a , b ]. After having heard the motivating example and the intuitive definition most students are able to formally define the density of U. U is such a variable that its density is a null outside [ a , b ] and b constant inside [ a , b ]. In the interval formulas 5. Using definitions 6. We have to find what is called a primitive function.

This together with 6. There is no need to try to guess or remember this result. Combining 6. Unit 6. One element of a family of similar distributions can be thought of as a function. The argument of the function is called a parameter. All families we have seen so far are either one- parametric or two-parametric. Let us look at the examples.

For example, the result from Unit 5. The exact meaning of this will be explained in Theorem 6. The density of a standard normal is symmetric about zero. We take this for granted. Even though the standard normal can take arbitrarily large values, the probability of z taking large values quickly declines. Therefore such integrals have been tabulated. Whatever density and table you deal with, try to express the integral through the area given in the table. Table 1 on p. In the other two cases the derivation is similar.

Standardization of a normal distribution gives a standard normal distribution. This follows from 6. The standard normal in this notation is N 0,1. We see that the family of normal distributions is two-parametric. Each of the parameters takes an infinite number of values, and it would be impossible to tabulate all normal distributions.

This is why we need the standardization procedure. The first statement, called a point estimate, entails too much uncertainty. The second estimate, called an interval estimate, is preferable. Confidence intervals can be two-tail and one-tail.

The interest in confidence intervals leads us to the next definition. We say that it converges to a random variable X in distribution if FX n x approaches F x for all real x. Corollary 6. Consider vectors X , Y and x , y X , Y is a pair of random variables and x , y is a pair of real numbers. Suppose X and Y have densities p X and pY , respectively.

With these definitions we can prove Property 3 from Unit 6. R R Constructing independent standard normal variables. We wish to have independent standard normal z1 , z2 both must have the same density pz t. In a similar way for any natural n we can construct n independent standard normal variables. Discuss the five properties of integrals that follow their definition: do summations have similar properties? How do you prove that a density is everywhere nonnegative?

How do you justify the definition of the mean of a continuous random variable? With the knowledge you have, which of the 18 properties listed in Unit 6. Define the uniformly distributed random variable. Find its mean, variance and distribution function.

In one block give the properties of the standard normal distribution, with proofs, where possible, and derive from them the properties of normal variables. Divide the statement of the central limit theorem into a preamble and convergence statement. How is this theorem applied to the binomial variable? Draw a parallel between the proofs of multiplicativity of means for discrete and continuous random variables. Minimum required: Exercises 6.

The random variable is income of a person incomes have frequencies. Its mean is the parameter of the population we are interested in. Observing the whole population is costly. Therefore we want to obtain a sample and make inference conclusions about the population from the sample information.

A simple random sample satisfies two conditions: i Every object has an equal probability of being selected and ii The objects are selected independently. If a list of objects is available, it can be satisfied using random numbers generated on the computer. Suppose we have a list of N objects to choose from. One example is sampling without replacement see section 5. Another is quota sampling. In quota sampling elements are sampled until certain quotas are filled.

For example, if the percentage of men in the country is 49 and the percentage of employed is 80, one might want to keep the same percentages in the sample. The problem with this type of sampling is dependence inclusion of a new element in the sample depends on the previously selected ones and impossibility to satisfy more than one quota simultaneously, unless by coincidence. T is constructed from sample observations X 1 , Therefore T is a random variable and, as any random variable, it has a distribution.

In the situation just described T is called a statistic and its distribution is called a sampling distribution. Working definition. A statistic is a random variable constructed from sample observations and designed to estimate a parameter of interest.

To recapitulate, it is a random variable and it depends on the sample data. In general, it is believed that for any population parameter there exists a statistic that estimates it. Controlling variances in sampling is important for at least three reasons. Firstly, in a production process of a certain product large deviations of its characteristics from technological requirements are undesirable. Secondly, the statistical procedure for comparing sample means from two populations requires knowledge of corresponding variances.

Thirdly, in survey sampling by reducing the sample variance we reduce the cost of collecting the sample. Exercise 7. Your solution may be very different from mine. From the proof we see that the variables must be i.

The beginning of the proof reminds me the derivation of the alternative expression for variance. This trick will prove useful again in Chapter Always try to see general features in particular facts. We apply the population mean to both sides of 7. Unit 7. Let z1 , Theorem 7. Try to perform more operations mentally. Tigran Petrosian became a chess grandmaster because at elementary school his teacher wanted him to write detailed solutions, while he wanted to give just an answer.

You might want to review the proof of Corollary 6. By Theorem 7. X takes only values 0 and 1. The mathematical definition 7.

If condition is true, it gives v1 , and if it is wrong, the command gives v2. Thus, Exercise 2. With three independent variables X 1 , X 2 , X 3 defined by 7.

When you put the function RAND in different cells, the resulting variables will be independent. Therefore Exercise 2. Let f be a function with the domain D f and range R f. In equations try to write the knowns on the right and the unknowns or what you define on the left side. Example 7. Thus, a function may not be invertible globally on the whole domain but can be invertible locally on a subset of a domain.

Suppose f is invertible. Then f is nondecreasing if and only if its inverse is nondecreasing. Let f be nondecreasing and let us prove that its inverse is nondecreasing.

Hence, the assumption is wrong and the inverse is nondecreasing. Condition sufficient for invertibility of a distribution function. The distribution function FX of a random variable X is invertible where the density p X is positive. Suppose p X is positive in some interval [ a, b] and let us show that the distribution function is invertible in that interval. Corollary 7. For U a ,b its distribution function is invertible on [ a, b]. Y is 0 identically distributed with X.

Despite being short, this proof combines several important ideas. Make sure you understand them. Hence, in Exercise 2. Fill out the next table: Table 7. How would you satisfy the definition of the simple random sampling in case of measuring average income of city residents? Define a sampling distribution and illustrate it with a sampling distribution of a sample proportion. Stating all the necessary assumptions, derive the formulas for the mean and variance of the sample mean.

Suppose we sample from a Bernoulli population. How do we figure out the approximate value of p? How can we be sure that the approximation is good? Define unbiasedness and prove that the sample variance is an unbiased estimator of the population variance. Find the mean and variance of the chi-square distribution. Find V s X2 and a confidence interval for sX2.

Minimum required: 7. In my exposition it was more appropriate to put that definition in Unit 7. Exercise 8. This exercise makes clear that even though unbiasedness is a desirable property, it is not enough to choose from many competing estimators of the same parameter.

In statistics we want not only to be able to hit the target on average but also to reduce the spread of estimates around their mean. This is a special type of a linear combination which is convenient to parameterize the points of the segment [T1 , T2 ]. In these two cases it is not possible to reduce variance by forming linear combinations.

In all other cases T a is more efficient than either T1 or T2. I give more information but am not sure that it will simplify the topic. Consistency is an asymptotic property and therefore we have to work with a sequence of estimators.

Let pn denote the density of Tn. Theorem 8. If we want to compare them, in both cases we have to assume an infinite sequence of estimators. We can ask what is the relationship between i consistency of Tn and ii unbiasedness of Tn for every n. An unbiased estimator sequence may have a large variance. When choosing between such a sequence and a consistent sequence we face a tradeoff: we can allow for a small bias if variance reduces significantly.

Unit 8. A sample of size n is drawn from a normal population. We want to prove that this is really a standard normal. From Theorem 8. And we know that, being a standardized variable, it has mean zero and variance 1. Thus, 8. A number of exercises consider individual analyses that are typically part of larger research projects. With this structure, students can deal with important detailed questions and can also work with case studies that require them to identify the detailed questions that are logically part of a larger research project.

These large data sets can also be used by the teacher to develop additional research and case study projects that are custom designed for local course environments. The opportunity to custom design new research questions for students is a unique part of this textbook. A complete list of the data files and where they are used is located at the end of this preface. Data files are also shown by chapter at the end of each chapter.

The book provides a complete and in-depth presentation of major applied topics. An initial read of the discussion and application examples enables a student to begin working on simple exercises, followed by challenging exercises that provide the opportunity to learn by doing relevant analysis applications. Chapters also include summary sections, which clearly present the key components of application tools.

Many analysts and teachers have used this book as a reference for reviewing specific applications. Once you have used this book to help learn statistical applications, you will also find it to be a useful resource as you use statistical analysis procedures in your future career. A number of special applications of major procedures are included in various sections. Clearly there are more than can be used in a single course. But careful selection of topics from the various chapters enables the teacher to design a course that provides for the specific needs of students in the local academic program.

Special examples that can be left out or included provide a breadth of opportunities. The initial probability chapter, Chapter 3, provides topics such as decision trees, overinvolvement ratios, and expanded coverage of Bayesian applications, any of which might provide important material for local courses.

Confidence interval and hypothesis tests include procedures for variances and for categorical and ordinal data. Random-variable chapters include linear combination of correlated random variables with applications to financial portfolios. Regression applications include estimation of beta ratios in finance, dummy variables in experimental design, nonlinear regression, and many more. As indicated here, the book has the capability of being used in a variety of courses that provide applications for a variety of academic programs.

The design of the book makes it possible for a student to come back to topics after several years and quickly renew his or her understanding. With all the additional special topics, that may not have been included in a first course, the book is a reference for learning important new applications. And the presentation of those new applications follows a presentation style and uses understandings that are familiar. This reduces the time required to master new application topics.

We understand how important it is for students to know statistical concepts and apply those to different situations they face everyday or will face as managers of the future. Almost all sections include examples that illustrate the application of the concepts or methods of that section to a real-world context even though the company or organization may be hypothetical.

Problems are structured to present the perspective of a decision maker and the analysis provided is to help understand the use of statistics in a practical way. The section exercises for each chapter begin with straightforward exercises targeted at the topics in each section. These are designed to check understanding of specific topics. Because they appear after each section, it is easy to turn back to the chapter to clarify a concept or review a method.

The Chapter Exercises and Applications are designed to lead to conclusions about the real world and are more application based. They usually combine concepts and methods from different sections. The authors thank Dr. Department of Agriculture, for her assistance in providing several major data files and for guidance in developing appropriate research questions for exercises and case studies.

We also thank Paula Dutko and Empharim Leibtag for providing an example of complex statistical analysis in the public sector. We also recognize the excellent work by Annie Puciloski in finding our errors and improving the professional quality of this book. In addition, we express special thanks for continuing support from our families. In addition, Betty acknowledges in memory the support of her parents, Westley and Jennie Moore.

The authors acknowledge the strong foundation and tradition created by the original author, Paul Newbold. Paul understood the importance of rigorous statistical analysis and its foundations. He realized that there are some complex ideas that need to be developed, and he worked to provide clear explanations of difficult ideas.

In addition, he realized that these ideas become useful only when used in realistic problem solving situations. Thus, many examples and many applied student exercises were included in the early editions. We have worked to continue and expand this tradition in preparing a book that meets the needs of future business leaders in the information age.

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