Probability Random Variables Chapter 8

Probability Random Variables Chapter 8 - CHAPTER STATISTICS...

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CHAPTER STATISTICS &1 INTRODUCTION Probability is a mathematical discipline developed as an abskact model and its con- clusions are deductionr based on tlte axioms. Statistics deals with the applications of the theory o real problems and its conclusions arc inferences based on observations. Statistics consists of two parts: analysis and design. Atwlysis, ot mathematical statistics, is part of probability involving mainly repeated trials and events the probability of which is close to 0 orto 1. This leads to inferences that can be accepted as near certainties (see pages ll-12). Design, or applied statistics, deals with data collection and construction of experiments that can be adequately described by probabilistic models. In this chapter, we introduce the basic elements of mathematical statistics. We start with the observation that the connection between probabilistic concepts and rcality is basedon the approximation (8-1) relatingtheprobabilityp: P(A)of aneventAtothenumbernAof successesof Ainn trials of the underlying physical experiment. We used this empirical formula to give the relative frequency interpretation of all probabilistic concepts. For example, we showed that the meatn n of a random variable x can be approximated by the average 4=1fri=r (8-2) of the observed values xi of x, and its distribution F(r) by the empirical distribution ^n- F(x) =: n (8-3) wtrere n, is the number of x;'s that do not exceed.r. These relationships are empirical 303 fl4 p=- n
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3ll4 pRoBABtury AND nlNmM vARIABrls Estimate 0 (b) FIGURE&1 point estimates of the parameters 4 and F(t) and a major objective of statistics is to give them an exact interPrctation. In a statistical investigation, we deal with two general classes of problems. In the first class, we assume that the probabilistic model is known and we wish to make predictions conceming future observations. For example, we know the distribution F(.r) -of a random variable x and we wish to predict the average f of its future samples or we know the probability p of an event A and we wish to predict the number na oi successes of A in z nrture Eials. In both cases, we proceed from the model to ttre observations (Fig. 8-14). In the second class, one or more iarameten 9; of the model are unknown and our objective is either to estinote their values (parameter estimation) or to dccidc,whether 0i is a set of known constants osi @ypothesis testing). For example, we observe the values ri of a random variable x and we wish to estimate its mean ; or to decide whether to accept the hypothesis that 4 : 5.3. We toss a coin 1000 times and heads shows 465 times. Using this information, we wish to estimate thq probability p of heads ortodecide whetherthe coinis fair.Inbothcases, we proceedfrom the observations to the model @g.
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Probability Random Variables Chapter 8 - CHAPTER STATISTICS...

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