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“数学很美，数学很有趣，数学很有竞争性，她是世界上最聪明的人玩的游戏。”刚刚获得2002年菲尔茨奖的符拉基米尔·费沃特斯基，对数学作出风趣的描述
你认为数学美吗？美在哪里？
数学的美感在于它的简单、和谐、丝丝入扣。就像古代描写美人：增一分则太肥，少一分则太瘦。数学就是这样的美人。在数学的世界里，有无穷的问题，人要有常青的思想，这真是一种享受。 
您认为学好数学对我们有哪些帮助？
数学是一门逻辑推导的学科，学好数学不全是为了当数学家，而是培养一种素质。人的逻辑推理能力，主要来自语言和数学。数学对科学家来说，本身就是一种语言，也是工具，学好数学就等于掌握了提高逻辑推理能力的一把金钥匙。” 
我国的数学教育专家在数学教育高级研讨班纪要中指出数学素质包括：广博的数学知识、准确的科学语言、良好的计算能力、周密的思维习惯、敏锐的数量意识，以及解决问题的数学技术。 
数学思维具有广泛的涵义，它除了具有明显的概括性、抽象性、逻辑性、精确性与定量性外，还具有问题性、相似类比性、辨证性、想象与猜测性以及直觉、美感等特性。 

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