數(shù)學(xué)論文翻譯-數(shù)學(xué)翻譯-數(shù)學(xué)翻譯公司
“數(shù)學(xué)很美，數(shù)學(xué)很有趣，數(shù)學(xué)很有競(jìng)爭(zhēng)性，她是世界上最聰明的人玩的游戲�！眲倓偒@得2002年菲爾茨獎(jiǎng)的符拉基米爾·費(fèi)沃特斯基，對(duì)數(shù)學(xué)作出風(fēng)趣的描述
你認(rèn)為數(shù)學(xué)美嗎？美在哪里？
數(shù)學(xué)的美感在于它的簡(jiǎn)單、和諧、絲絲入扣。就像古代描寫美人：增一分則太肥，少一分則太瘦。數(shù)學(xué)就是這樣的美人。在數(shù)學(xué)的世界里，有無(wú)窮的問(wèn)題，人要有常青的思想，這真是一種享受。
您認(rèn)為學(xué)好數(shù)學(xué)對(duì)我們有哪些幫助？
數(shù)學(xué)是一門邏輯推導(dǎo)的學(xué)科，學(xué)好數(shù)學(xué)不全是為了當(dāng)數(shù)學(xué)家，而是培養(yǎng)一種素質(zhì)。人的邏輯推理能力，主要來(lái)自語(yǔ)言和數(shù)學(xué)。數(shù)學(xué)對(duì)科學(xué)家來(lái)說(shuō)，本身就是一種語(yǔ)言，也是工具，學(xué)好數(shù)學(xué)就等于掌握了提高邏輯推理能力的一把金鑰匙。”
我國(guó)的數(shù)學(xué)教育專家在數(shù)學(xué)教育高級(jí)研討班紀(jì)要中指出數(shù)學(xué)素質(zhì)包括：廣博的數(shù)學(xué)知識(shí)、準(zhǔn)確的科學(xué)語(yǔ)言、良好的計(jì)算能力、周密的思維習(xí)慣、敏銳的數(shù)量意識(shí)，以及解決問(wèn)題的數(shù)學(xué)技術(shù)。
數(shù)學(xué)思維具有廣泛的涵義，它除了具有明顯的概括性、抽象性、邏輯性、精確性與定量性外，還具有問(wèn)題性、相似類比性、辨證性、想象與猜測(cè)性以及直覺(jué)、美感等特性。
Suppose you test one circuit. With probability p, the circuit is rejected. Let X be the number of rejected circuits in one test. What is ?
Because there are only two outcomes in the sample space, X = 1 with probability p and X = 0 with probability .
Therefore, the number of circuits rejected in one test is a Bernoulli random variable.
Example 2.10 If there is a 0.2 probability of a reject.
Example 2.11 In a test of integrated circuits there is a probability p that each circuit is rejected. Let Y equal the number of tests up to and including the first test that discovers a reject. What is the PMF of Y?
The experiement is simply to keep testing circuits until a reject appears. Using a denote an accepted circuit and r to denote a reject, the tree is
From the tree, we see that P[Y =1] = p, P[Y = 2] = p( ), P[Y = 3] = , and in general, P[Y = y] = . Therefore,
Y is a referred to as a geometric random variable because the probabilites in the PMF decline geometrically.
Definition 2.6. Geometric Random Variable: X is a Geometric random variable if the PMF of X has the form
Where the parameter p is in the range 0 < p < 1.
Example 2.12 If there is a 0.2 probability of a reject.
Example 2.13 Suppose we test n circuits and each circuit is rejected with probability p independent of the results of other tests. Let K equal the number of rejects in the n tests. Find the PMF .
Adopting the vocabulary of Section 1.9, we call each discovery of a defective circuit a success and each test is an independent trial with success probability p. The event K = k corresponds to k successes in n trials, which we have already found, in Equation (1.6), to be the binomial probability
K is an example of a binomial random variable.
Definition 2.7 Binomial Random Variable: X is binomial random variable if the PMF of X has the form
Where 0 < p < 1 and n is an integer such that .
Whenever we have a sequence of n independent trials each with success probability p, the number of successes is a binomial random variable. In general, for a binomial random variable with parameters n and p, we call n the number of trials and p the success probability. Note that a Bernoulli random variables is a binomial random variable with n = 1.
Example 2.14 If there is a 0.2 probability of a reject and we perform 10 tests.
Example 2.15 Suppose you test circuits until you find k rejects. Let L equal the number of tests. What is the PMF of L?
For large value of k, the tree become difficult to draw. Once again, we view the tests as a sequenc |
