Mathematical English 12: Probability Theory and Mathematical StatisticsDr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 1Mathematical EnglishDr. Xiaomin ZhangEmail: zhangxiaominnbu.edu.cnMathematical English 12: Probability Theory and Mathematical StatisticsDr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 22.12 Probability Theory and Mathematical StatisticsTEXT A Special terminology peculiar to probability theoryIn discussions involving probability, one often sees phrases from everyday language such as “two events are equally likely,” “an event is impossible,” or “an event is certain to occur.” Expressions of this sort have intuitive appeal and it is both pleasant and helpful to be able to employ such colorful language in mathematical discussions. Before we can do so, however, it is necessary to explain the meaning of this language in terms of the fundamental concepts of our theory.Because of the way probability is used in practice, it is convenient to imagine that each probability space (S, B, P) is associated with a real or conceptual experiment. The universal set S can then be thought of as the collection of all conceivable outcomes of the experiment, as in Mathematical English 12: Probability Theory and Mathematical StatisticsDr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 3the example of coin tossing discussed in the foregoing section. Each element of S is called an outcome or a sample and the subsets of S that occur in the Boolean algebra B are called events. The reasons for this terminology will become more apparent when we treat some examples.Assume we have a probability space (S, B, P) associated with an experiment. Let A be an event, and suppose the experiment is performed and that its outcome is x. (In other words, let x be a point of S.) This outcome x may or may not belong to the set A. If it does, we say that the event A has occurred. Otherwise, we say that the event A has not occurred, in which case xA, so the complementary event A has occurred. An event A is called impossible if A=, because in this case no outcome of the experiment can be an element of A. The event A is said to be certain if A=S, because then every outcome is automatically an element of A.Mathematical English 12: Probability Theory and Mathematical StatisticsDr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 4Each event A has a probability P(A) assigned to it by the probability function P. (The actual value of P(A) or the manner in which P(A) is assigned is not concern us at present.) The number P(A) is also called the probability that an outcome of the experiment is one of the elements of A. We also say that P(A) is the probability that the event A occurs when the experiment is performed.The impossible event must be assigned probability zero because P is finitely additive measure. However, there may be events with probability zero that are not impossible. In other words, some of the nonempty subsets of S may be assigned probability zero. The certain event S must be assigned probability 1 by the very definition of probability, but there may be other subsets as well that are assigned probability 1. In example 1 of Section 6.8 there are nonempty subsets with probability zero and proper subsets of S that have probability 1.Two events A and B are said to be equally likely if P(A)=P(B). The Mathematical English 12: Probability Theory and Mathematical StatisticsDr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 5event A is called more likely than B if P(A)P(B), and at least as likely as B if P(A)P(B). Table 2-12-1 provides a glossary or further everyday language that is often used in probability discussions. The letters A and B represent events, and x represents an outcome of an experiment associated with the sample space S. Each entry in the left-hand column is a statement about the events A and B, and the corresponding entry in the right-hand column defines the statement in terms of set theory.Mathematical English 12: Probability Theory and Mathematical StatisticsDr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 6Notationsprobability function here the value of probability function P at point A is the probability that the event A occurs. Generally, The probability function P(x) (also called the probability density function or density function) of a continuous distribution is defined as the derivative of the (cumulative) distribution function D(x), so A probability function satisfies Mathematical English 12: Probability Theory and Mathematical StatisticsDr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 7and is constrained by the normalization condition, Special cases are To find the probability function in a set of transformed variables, find the Jacobian. For example, If u=u(x), the