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数学代写|概率论代考Probability Theory代写|The sample space

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数学代写|概率论代考Probability Theory代写|The sample space

R. von Mises introduced the idea of a sample space in $1931^4$ and while his frequency-based ideas of probability did not gain traction-and were soon to be overtaken by Kolmogorov’s axiomatisation – the identification of the abstract sample space of a model experiment paved the way for the modern theory.
We shall denote by the uppercase Greek letter $\Omega$ an abstract sample space. It represents for us the collection of idealised outcomes of $\mathrm{a}$, perhaps conceptual, chance experiment. The elements $\omega$ of $\Omega$ will be called sample points, each sample point $\omega$ identified with an idealised outcome of the underlying gedanken experiment. The sample points are the primitives or undefined notions of the abstract setting. They play the same rôle in probability as the abstract concepts of points and lines do in geometry.

The simplest setting for probability experiments arises when the possible outcomes can be enumerated, that is to say, the outcomes are either finite in number or denumerably infinite. In such cases the sample space is said to be discrete. The examples of the previous section all deal with discrete spaces.
ExAmples: 1) A coin toss. The simplest non-trivial chance experiment. The sample space consists of two sample points that we may denote $\mathfrak{H}$ and $\mathfrak{T}$.
2) Three tosses of a coin. The sample space corresponding to the experiment of Example 2.1 may be represented by the aggregate $\mathfrak{H} \mathfrak{H} \mathfrak{H}, \mathfrak{H} \mathfrak{H} \mathfrak{T}, \ldots, \mathfrak{T} \mathfrak{T} \mathfrak{T}$ of eight sample points.
3) A throw of a pair of dice. The sample space consists of the pairs $(1,1),(1,2), \ldots$, $(6,6)$ and has 36 sample points. Alternatively, for the purposes of Example 2.2 we may work with the sample space of 11 elements comprised of the numbers 2 through 12.
4) Hands at poker, bridge. A standard pack of cards contains 52 cards in four suits (called spades, hearts, diamonds, and clubs), each suit containing 13 distinct cards labelled 2 through 10 , jack, queen, king, and ace, ordered in increasing rank from low to high. In bridge an ace is high card in a suit; in poker an ace counts either as high (after king) or as low (before 2). A poker hand is a selection of five cards at random from the pack, the sample space consisting of all $\left(\begin{array}{c}52 \ 5\end{array}\right)$ ways of accomplishing this. A hand at bridge consists of the distribution of the 52 cards to four players, 13 cards per player. From a formal point of view a bridge hand is obtained by randomly partitioning a 52-card pack into four equal groups; the sample space of bridge hands hence consists of $(52) ! /(13 !)^4$ sample points. In both poker and bridge, the number of hands is so large that repetitions are highly unlikely; the fresh challenge that each game presents contributes no doubt in part to the enduring popularity of these games.
5) The placement of two balls in three urns. The sample space corresponding to Example 2.3 may be represented by the aggregate of points (2.1).
6) The selection of a random graph on three vertices. A graph on three vertices may be represented visually by three points (or vertices) on the plane potentially connected pairwise by lines (or edges). There are eight distinct graphs on three vertices-one graph with no edges, three graphs with one edge, three graphs with two edges, and one graph with three edges-each of these graphs constitutes a distinct sample point. A random graph (traditionally represented $\mathrm{G}_3$ instead of $\omega$ in this context) is the outcome of a chance experiment which selects one of the eight possible graphs at random. Random graphs are used to model networks in a variety of areas such as telecommunications, transportation, computation, and epidemiology.
7) The toss of a coin until two successive outcomes are the same. The sample space is denumerably infinite and is tabulated in Example 2.4. Experiments of this stripe provide natural models for waiting times for phenomena such as the arrival of a customer, the emission of a particle, or an uptick in a stock portfolio.

While probabilistic flavour is enhanced by the nature of the application at hand, coins, dice, graphs, cards, and so on, the theory of chance itself is independent of semantics and the specific meaning we attach in a given application to a particular outcome. Thus, for instance, from the formal point of view we could just as well view heads and tails in a coin toss as 1 and 0 , respectively, without in any material way affecting the probabilistic statements that result. We may choose hence to focus on the abstract setting of discrete experiments by simply enumerating sample points in any of the standard ways (though tradition compels us to use the standard notation for these spaces instead of $\Omega$ ).
EXAMPLES: 8) The natural numbers $\mathbb{N}$. The basic denumerably infinite sample space consists of the natural numbers $1,2,3, \ldots$
9) The integers $\mathbb{Z}$. Another denumerably infinite sample space consisting of integer-valued sample points $0, \pm 1, \pm 2, \ldots$

数学代写|概率论代考Probability Theory代写|Sets and operations on sets

An abstract sample space $\Omega$ is an aggregate (or set) of sample points $\omega$. We identify events of interest with subsets of the space at hand. To describe events and their interactions we hence resort to the language and conventions of set theory. We begin with a review of notation and basic concepts.

Accordingly, suppose $\Omega$ is an abstract universal set. A subset $A$ of $\Omega$ is a subcollection of the elements of $\Omega$. As is usual, we may specify the subsets of $\Omega$ by membership, $A={\omega: \omega$ satisfies a given property $\mathcal{P}}$, by an explicit listing of elements, $A=\left{\omega_1, \omega_2, \ldots\right}$, or, indirectly, in terms of other subsets via set operations as we detail below. If $\omega$ is an element of $A$ we write $\omega \in A$.
We reserve the special symbol $\varnothing$ for the empty set containing no elements.

Suppose the sets $A$ and $B$ are subcollections of elements of $\Omega$. We say that $A$ is contained in $B$ (or $A$ is a subset of $B$ ) if every element of $A$ is contained in $B$ and write $A \subseteq B$ or $B \supseteq A$, both notations coming to the same thing. By convention, the empty set is supposed to be contained in every set. Two sets $A$ and $B$ are equivalent, written $A=B$, if, and only if, $A \subseteq B$ and $B \subseteq A$. To verify set equality $A=B$ one must verify both inclusions: first show that any element $\omega$ in $A$ must also be in $B$ (thus establishing $A \subseteq B$ ) and then show that any element $\omega$ in $B$ must also be in $A$ (thus establishing $B \subseteq A$ ). Finally, the sets $A$ and $B$ are disjoint if they have no elements in common.

Given sets $A$ and $B$, new sets may be constructed by disjunctions, conjunctions, and set differences. The union of $A$ and $B$, written $A \cup B$, is the set whose elements are contained in $A$ or in $B$ (or in both). The intersection of $A$ and $B$, written $A \cap B$, is the set whose elements are contained both in $A$ and in $B$. The set difference $A \backslash B$ is the set whose members are those elements of $A$ that are not contained in $B$; the special set difference $\Omega \backslash A$ is called the complement of $A$ and denoted $A^c$. Finally, the symmetric difference between $A$ and $B$, denoted $A \triangle B$, is the set of points that are contained either in $A$ or in $B$, but not in both. These operations may be visualised in Venn diagrams as shown in Figure 1.

概率论代写

数学代写|概率论代考Probability Theory代写|The sample space

R. von Mises 在 $1931^4$ 虽然他基于频率的概率思想没有获得关注一一并且很快被 Kolmogorov 的公理化所取代一一但模型实验的 抽象样本空间的识别为现代理论俌平了遉路。

2）掷三次硬币。示例2.1的实验对应的样本空间可以用聚合表示 $\mathfrak{H} \mathfrak{H} \mathfrak{H}, \mathfrak{H} \mathfrak{H} \mathfrak{T}, \ldots, \mathfrak{T} \mathfrak{T} \mathfrak{T}$ 八个样本点。
3) 㧷一对骰子。样本空间由对组成 $(1,1),(1,2), \ldots,(6,6)$ 并有 36 个采样点。或者，为了示例 2.2 的目的，我们可以使用由数 字 2 到 12 组成的 11 个元嗉的样本空间。
4) 手牌、桥牌。一副标准的纸牌包含 52 张牌，分为四种花色（称为黑桃、红心、方块和梅花），每种花色包含 13 张不同的牌， 分别标记为 2 到 $10 、 J 、 Q 、 K$ 和 $A$ ，排列顺序从低到高高的。在桥牌中， $A$ 是花色中的高牌; 在扑克中， $A$ 要么算高 (在 $K$ 之 后)，要么算低 (在 2 之前)。一手牌是从一副牌中随机选择五张牌，样本空间由所有牌组成 ( 525 )实现这一点的方法。枡牌手 牌包括将 52 张牌分配给四名玩家，每名玩家 13 张牌。从正式的角度来看，柇牌是通过将 52 张牌随机分成四个相等的组来获得 的; 因此㭞手的样本空间包括 $(52) ! /(13 !)^4$ 样本点。在扑克和秎牌中，手牌的数量非常多，重㫜的可能生很小; 每款游戏所带来 的新鲜挑战无疑在一定程度上促成了这些斿戏的持久流行。
5) 将两个球放入三个缸中。示例 2.3 对应的样本空间可以由点 (2.1) 的焦合表示。
6) 三个顶点上随机图的选择。三个顶点上的图可以在视觉上由平面上的三个点 (或顶点) 表示，平面上可能由线 (或边) 成对连 接。在三个顶点上有八个不同的图一一一个没有边的图、三个有一条边的图、三个有两条边的图和一个有三条边的图一一这些图中 的每一个构成一个不同的样本点。随机图 (传统上表示 $\mathrm{G}_3$ 代替 $\omega$ 在这种情况下) 是随机选择八个可能图形之一的机会实验的结 果。随机图用于对电信、交通、计算和流行病学等各个领域的网絡建模。
7) 抛硬币直到两个连续的结果相同。样本空间是可数无限的，如例 2.4 所示。这条条纹的实验为客户到达、粒子发射或股票投资 组合上张等现彖的等待时间提供了自然模型。

9) 整数 $\mathbb{Z}$. 另一个由整数值样本点组成的可数无限样本空间 $0, \pm 1, \pm 2, \ldots$

数学代写|概率论代考Probability Theory代写|Sets and operations on sets

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MATLAB代写

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