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This book is written in four major divisions. The first part is the introductory chapters consisting of Chapters 1 and 2. In part two, Chapters 3–11, we develop fuzzy estimation. For example, in Chapter 3 we construct a fuzzy estimator for the mean of a normal distribution assuming the variance is known. More details on fuzzy estimation are in Chapter 3 and then after Chapter 3, Chapters 4–11 can...
In this chapter we have collected together the basic ideas from fuzzy sets and fuzzy functions needed for the book. Any reader familiar with fuzzy sets, fuzzy numbers, the extension principle, α-cuts, interval arithmetic, and fuzzy functions may go on and have a look at Section 2.5. In Section 2.5 we discuss the method we will be using in this book to evaluate comparisons between fuzzy numbers. That...
This starts a series of chapters, Chapters 3–11, on fuzzy estimation. In this chapter we first present some general information on fuzzy estimation and then concentrate on the mean of a normal probability distribution assuming the variance is known. The rest of the chapters on fuzzy estimation can be read independently.
Consider X a random variable with probability density function N(μ, σ2), which is the normal probability density with unknown mean μ and unknown variance σ2. To estimate μ we obtain a random sample X1, ..., Xn from N(μ, σ2). Suppose the mean of this random sample turns out to be 4.2 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn...
We have an experiment in mind in which we are interested in only two possible outcomes labeled “success” and “failure”. Let p be the probability of a success so that q = 1−p will be the probability of a failure. We want to estimate the value of p. We therefore gather a random sample which here is running the experiment n independent times and counting the number of times we had a success. Let x be...
We first construct a fuzzy estimator for σ2 using the usual confidence intervals for the variance from a normal distribution and we show this fuzzy estimator is biased. Then in Section 6.3 we construct an unbiased fuzzy estimator for the variance.
We have two populations: Pop I and Pop II. Pop I is normally distributed with unknown mean µ1 and known variance σ12 .Pop II is also normally distributed with unknown mean µ2 but known variance σ22 . We wish to construct a fuzzy estimator for µ1 − µ2.
This continues Chapter 7 but now the population variances are unknown. We will use the notation of Chapter 7. There are three cases to look at and we will discuss each of these below.
Let x1,..., xn be the values of a random sample from a population Pop I. Let objecti, or personi, belong to Pop I which produced measurement xi in the random sample, 1 ≤ i ≤ n. Then, at possibly some later time, we take a second measurement on object...
In this chapter we have two binomial population Pop I and Pop II. In Pop I (II) let p1 (p2) be the probability of a “success”. We want a fuzzy estimator for p1 — p2.
We first discuss the non-fuzzy confidence intervals for the ratio of the variances. We then construct a fuzzy estimator for σ12/σ22 from these crisp confidence intervals and show it is biased (see Chapter 6). Then we find an unbiased fuzzy estimator.
This chapter starts a series of chapters (Chapters 12–20) on fuzzy hypothesis testing. In each chapter we first review the crisp case first before proceeding to the fuzzy situation. We give more details on fuzzy hypothesis testing in this chapter.
We have two populations: Pop I and Pop II. Pop I is normally distributed with unknown mean μ1 and unknown variance σ12 . Pop II is also normally distributed with unknown mean μ2 and unknown variance σ22 . We wish to do the following statistical test 17.1 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn...
In this chapter we have two binomial populations: Pop I and Pop II. In Pop I (II) let p1 (p2) be the probability of a “success”. We want a do the following test 18.1 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs...
We begin following Chapter 9. Let x1, ..., xn be the values of a random sample from a population Pop I. Let objecti, or personi,belong to Pop I which produced measurement xi in the random sample, 1 ≤ i ≤ n. Then, at possibly some later time, we take a second measurement...
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