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Show that the probability distribution for X = total on the two dice is as shown in the figure hiv infection from hospital purchase amantadine pills in toronto. For example best antiviral juice purchase amantadine 100mg online, you could denote the six outcomes where you get a 1 on the first die by (1 hiv infection time period buy amantadine 100 mg with visa, 1) antiviral used for cold sores order amantadine amex, (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), where the second number in a pair is the number you get on the second die. Show that the probabilities in part b satisfy the two conditions for a probability distribution. Hint: For each outcome in the sample space, indicate the value of X assigned to that outcome. Probability Distribution of X = Maximum Number for Rolls of Two Dice x 1 2 3 4 5 6 P(x) 1/36 3/36 5/36 7/36 9/36 11/36 275 6. Show that the two conditions in the definition of a probability distribution are satisfied. San Francisco Giants hitting the table shows the probability distribution of the number of bases for a randomly selected time at bat for a San Francisco Giants player in 2010 (excluding times when the player got on base because of a walk or being hit by a pitch). Interpret the mean, explaining why it does not have to be a whole number, even though each possible value for the number of bases is a whole number. Specify the probability distribution for the grade point score of a randomly selected student of this instructor. Selling houses Let X represent the number of homes a real estate agent sells during a given month. Explain why it does not make sense to compute the mean of this probability distribution as (0 + 1 + 2 + 3 + 4)/5 = 2. One is the Pick 3 game in which you pick one of the 1000 three-digit numbers between 000 and 999. With a bet of $1, you win $500 if your number is selected and nothing ($0) otherwise. In terms of your expected winnings, with which game are you better off-playing Pick 4, or playing Pick 3 in which you win $500 for a correct choice of a three-digit number? You are given $10 and told that you must pick one of two wagers, for an outcome based on a spin of the wheel: (1) Bet $10 on number 23. If the spin results in any one of the black numbers, you win $10 and also get back your $10 bet. Gambling some more Consider a game of poker being played with a standard 52-card deck (four suits, each of which has 13 different denominations of cards). If not, you will lose the hand as well as the $20 you called in order to receive the final card. On the other hand, if you do not call the bet, the hand ends immediately, your opponent wins, and you neither win nor lose any more money. Based on the expected value, should you call the $20 bet and receive one more card or not call the bet? Probability Distribution of X = Ideal Number of Children x 0 1 2 3 4 P(x) Females 0. Explain why a practical implication of the values for the standard deviations is that males hold slightly more consistent views than females about the ideal family size. Suppose the farmer buys insurance for $3000 that pays him $20,000 in the event of a severe storm that destroys much of the crop and pays nothing otherwise. An Associated Press story (April 28, 2007) about this software described a case in which a retail chain sold three similar power drills: one for $90, a better one for $120, and a top-tier one for $130. Software predicted that by selling the middle-priced drill for only $110, the cheaper drill would seem less a bargain and more people would buy the middle-price drill. For the original pricing, suppose 50% of sales were for the $90 drill, 20% for the $120 drill, and 30% for the $130 drill. Construct the probability distribution of X = selling price for the sale of a drill, and find its mean and interpret, b. For the new pricing, suppose 30% of sales were for the $90 drill, 40% for the $110 drill, and 30% for the $130 drill. Is the mean of the probability distribution of selling price higher with this new pricing strategy? Sketch a curve of the probability distribution of this random variable, which is the continuous version of the uniform distribution (see Exercise 6. We next learn about a probability distribution, called the normal distribution, that is commonly used for continuous random variables. It is characterized by a particular5 symmetric, bell-shaped curve with two parameters- the mean and the standard deviation.
Of these we are mainly interested in the error arising from the reduction antiviral research impact factor 2015 order cheapest amantadine and amantadine, since this is the part of the process which has been most criticized hiv infection latent stage cheap amantadine 100 mg without prescription. We adopt the description of the process given in §4 hiv infection rate zimbabwe buy 100mg amantadine overnight delivery, note (1) hiv-1 infection cycle order amantadine 100 mg with amex, and observe that apart from a slight difference in the form of the matrices Jr, the reduction is similar to the Jordan process. Assuming there are no back-substitution errors, and ignoring the second-order terms in the errors we should have: n error in x = U-1 Xn r=1 X-1 (sr - Sr U-1 Xn b), r where Xr = Jr. Now, assuming that the process has been done with the largest pivot chosen from each column, we shall have M(X-1) = 1, for X-1 = 1 + (1 - Js) as mentioned in §4(2). If we arrange that M tr dr, the 2 M(A-1), and normally much smaller than the greatest error in any coefficient from this source is n error arising from the first part of the process. It is interesting to note the value of the error in the last pivot, that is, the error in the (nn) coefficient of Jn. The(nn) coefficient is dn r r anj (X-1 Sr)jn and since M(X-1) = 1, M(Sr) it r r Rounding-off Errors in Matrix Processes 401 does not exceed n2 dn M(A-1) in absolute magnitude, that is, the proportionate error in the last pivot is at most n2 M(A-1). J1 A may be written L-1 X-1 Sr, from which we find that the error in the last pivot cannot r r exceed n2 M(L-1). But since M(L) = 1 we find M(L-1) 2n-1 (and equality can be attained): that this error may actually be as great as 2n-2 may be seen by considering the inversion of a matrix differing only slightly from 1 0 0. Normally, even with ill-conditioned equations we may expect the off-diagonal coefficients of L to be distributed fairly uniformly between -1 and 1, possibly with a tendency to be near 0. Only when there is a strong tendency for negative values will we find a large L-1. Errors in the unsymmetrical Choleski method When obtaining the triangular resolution of a matrix by the method of the theorem (§3) it is convenient to think of the process as follows. Regarding the process in this way suggests the following rule for deciding the number of figures to be retained. We always retain sufficient figures to give us an error of not more than in the coefficient of A under consideration. All the figures in the multiplier register are then written down as the value of l94. When inverting W (say) we can arrange, by an exactly similar computing procedure, that its product with its reputed inverse differs from unity by at most in each coefficient, i. This paper is published with the permission of the Director of the National Physical Laboratory. Nor did Turing leave any drafts, letters, journals of his ideas or accounts of the development of his thought. There is one small exception to this otherwise complete lack of manuscript record. For in that year, his friend and fellow mathematics undergraduate David Champernowne made (and published) the observation that the number 0. In any case it is notable that the chosen title for his great paper put numbers first, and the application to logic second. This connection illustrates a wider point, which is that Turing in 1935 was not a specialised logician, but the product of a very strong Cambridge background in both pure and applied mathematics, who in 1934 had devoted himself to a proof of the Central Limit Theorem. The rigorous analysis of the real number system, as the foundation of continuous mathematics, was a central part of his knowledge. The difficulty posed by the non-unique decimal representation of the reals seems to have stopped this project in its tracks. In (Turing 1950) he used a pseudo-random sequence (the digits of) to illustrate the computer simulation of randomness, without any discussion of the concept. This is even odder when one recalls that he had spent 6 years of war on the work of distinguishing random from pseudo-random. I propose to show how normal numbers may be constructed and to prove that almost all numbers are normal constructively. If is any sequence of figures in that scale we denote by N(t, n, R) the number of these in which occurs exactly n times.
Knot theory does indeed have beautiful ap plications in contemporary physics- as Witten and others have shown- but this has nothing to do with Lacan hiv infection neutropenia 100mg amantadine with visa. The last paragraph plays on the postmodern fondness for "multidimensionality" and "nonlinearity" by inventing a nonex istent field: "multidimensional (nonlinear) logic" hiv infection chart purchase 100 mg amantadine overnight delivery. Note 73 is deliberately exaggerated hiv infection urine cheap amantadine 100mg mastercard, though we are sympa thetic to the idea that economic and political power struggles strongly affect how science gets translated into technology and for whose benefit antiviral eye ointment buy on line amantadine. Cryptography does indeed have military (as well as commercial) applications and has in recent years be come increasingly based on number theory. However, number theory has fascinated mathematicians since antiquity, and until recently it had very few "practical" applications of any kind: it was the branch of pure mathematics par excellence. The refer ence to Hardy was dangerous: in this very accessible autobiog raphy, he prides himself on working in mathematical fields that have no applications. Nev ertheless, it also contains some ideas- on the link between scientists and the military, on ideological bias in science, on the pedagogy o f science- with which we partly agree, at least when these ideas are formulated more carefully. We do not want the parody to provoke unqualified derision toward these ideas, and we refer the reader to the Epilogue for our real views on some of them. This section begins by claiming that "postmodern" science has freed itself from objective truth. But, whatever opinions sci1 "Manifold" is a geometrical concept that generalizes the notion o f surface to spaces " o f more than two dimensions. Nevertheless, a whole book would be needed to disen tangle the confusions concerning chaos, quantum physics, and self-organization that underlie this sort of idea; see Chapter 7 for a brief analysis. He has probably confused it with the recent, and very speculative, theories on complexity. Ditto for the relation between the axiom of choice15and the move 1 ment for abortion rights. Cohen has indeed shown that neither the axiom of choice nor its negation can be deduced from the other axioms of set theory; but this mathematical result has no political implications whatsoever. Finally, all the bibliographic entries are rigorously exact, apart from a wink at former French minister of culture Jacques Toubon, who tried to impose the use of French in scientific con ferences sponsored by the French government (see Kontsevitch 1994), and at Catalan nationalism (see Smolin 1992). Transgressing the Boundaries: An Afterword* L es grandes person nes son t d ecid em en t bien bizarres, se dit le p e tit prince. Clearly I owe the editors and readers of Social Text, as well as the wider intellectual com munity, a non-parodic explanation of my motives and my true views. Like the genre it is meant to satirize- myriad exemplars of which can be found in my reference list- my article is a melange of truths, half-truths, quarter-truths, falsehoods, non ·This article was submitted to Social Text following the publication o f the parody, but was rejected on the grounds that it did not meet their intellectual standards. See also the critical comment by Social Text co-founder Stanley Aronowitz (1997) and the reply by Sokal (1997b). In particular, the fact that I have parodied an extreme or ambiguously stated version o f an idea does not exclude that I may agree with a more nuaneed or precisely stated version o f the same idea. All works cited in my article are real, and all quotations are rigor ously accurate; none are invented. Rather, my concern is explicitly political: to combat a currently fash ionable postmodemist/poststructuralist/social-constructivist discourse- and more generally a penchant for subjectivism- 2 For example: "linear", "nonlinear", "local", "global", "multidimensional", "relative", "frame o f reference", "field", "anomaly", "chaos", "catastrophe", "logic", "irrational", "imaginary", "complex", "real", "equality", "choice". I am aware that this wisecrack is unfair to the more sophisticated relativist philosophers o f science, who will concede that em pirica l statements can be objectively true-. I think that also this view is largely wrong, but that is a much longer discussion. But Am erican de partments o f literature, history and so cio lo g y contain large numbers o f self-described leftists w h o have confused radical doubts about o b jectivity with p olitical radicalism, and are in a mess. But there "The natural sciences have little to fear, at least in the short run, from postmodernist silliness; it is, above all, history and the social sciences- and leftist politics- that suffer when verbal game-playing displaces the rigorous analysis o f social realities. Nevertheless, because o f the limitations o f my own expertise, my analysis here will be restricted to the natural sciences (and indeed primarily to the physical sciences). While the basic epistemology of inquiry ought to be roughly the same for the natural and social sciences, I am of course perfectly aware that many special (and very difficult) methodological issues arise in the social sciences from the fact that the objects o f inquiry are human beings (including their subjective states of mind); that these objects o f inquiry have intentions (including in some cases the concealment of evidence or the placement o f deliberately self-serving evidence); that the evidence is expressed (usually) in human language whose meaning may be ambiguous; that the meaning of conceptual categories. So by no means do I claim that my comments about physics should apply directly to history and the social sciences- that would be absurd.
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