### 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|Distribution of balls in boxes

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## 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|Distribution of balls in boxes

Let there be $r$ balls and $n$ boxes, which are numerated by the numbers $i=1,2, \ldots, n$. Denote the set of boxes by $\Omega_{0}={1,2, \ldots, n}$.

Let us first consider the case of distinguishable (i.e., having some differences from each other – number, color, etc.) balls.

Denote by $\Omega$ a sample space, corresponding to a random distribution of $r$ balls into $n$ boxes (here and further «random distribution of balls in boxes” means that any ball can get into any box with the same probability). If we denote by $i_{j}(j=1,2, \ldots, r)$ the number of box into which the ball No, $j$ got, then the sample space corresponding to the given experiment can be described as follows:
$$\Omega=\left{\left(i_{1}, i_{2}, \ldots, i_{r}\right): i_{j} \in \Omega_{0}, j=1,2, \ldots, r\right}=\underbrace{\Omega_{0} \times \Omega_{0} \times \ldots \times \Omega_{0}}_{r}$$
From this we see that the experiment consisting in placing $r$ distinguishable balls into $n$ distinguishable boxes and the experiment corresponding to the choice of a random sample of size $r$ from the general population of size $n$ are described by the same sample space (see the previous paragraph $1.1$ ).

Remark. Above we used the figurative language of «balls» and «boxes», but the sample space, constructed earlier for this scheme, allows a large number of interpretations.

For the convenience of further references, we present now a number of schemes that are visually very different but essentially equivalent to the abstract arrangement of $r$ balls in $n$ boxes in the sense that the corresponding outcomes differ only in their verbal description. In this case, the probabilities attributed to elementary events can be different in different examples.

Example 5. a) Birthdays. The distribution of birthdays of $r$ students corresponds to the distribution of $r$ balls into $n=365$ boxes (it is assumed that there are 365 days in a year).
b) When firing at targets, the bullet corresponds to the balls, and the targets to the boxes.
c) In experiments with cosmic rays, particles that fall into Geiger counters play the role of balls, and the counters themselves are boxes.
d) The elevator leaves (rises) with $r$ people and stops on $n$ floors. Then the distribution of people into groups, depending on the floor on which they exit, corresponds to the distribution of $r$ balls in $n$ boxes.
e) The experiment consisting in throwing $r$ dice corresponds to the distribution of $r$ balls in $n=6$ boxes. If the experiment consists in throwing $r$ symmetrical coins, then $n=2$.

From the above formula (17), according to Theorem 2 of the preceding section, it follows that $|\Omega|=n^{r}$. The latter means that $r$ distinguishable balls can be distributed over $n$ distinguishable boxes in $n^{r}$ ways.

In many cases it is necessary to consider the balls indistinguishable (the balls are the same and they do not differ from each other in color, shape, weight, etc.). For example, when examining the distribution of birthdays by days of the year, only the number of people born on a particular day is of interest (the number of balls that have fallen into a particular box).

To show that depending on whether the balls are distinguishable or indistinguishable, the number of possible balls distributions in the boxes may be different.
Let us give an example.

## 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|Statistics of Maxwell-Boltzmann

We consider several important distribution problems arising in the study of certain particle systems in physics and statistical mechanics.

In statistical mechanics, phase space is usually divided into a large number $n$ of small regions or cells so that each particle is assigned to one cell. As a result, the state of the whole system is described as a random arrangement of $r$ particles (balls) in $n$ cells (boxes).

The Maxwell-Boltzmann system is characterized as a system of $r$ distinguishable (different) particles, each of which can be in one of $n$ cells (states), regardless of where the remaining particles are located. In such a system it is possible to have $n^{r}$ different arrangements of $r$ particles into $n$ cells. If, in doing so, all such arrangements (states of the system) are considered equally probable, then we speak of MaxwellBoltzmann statistics. Thus, in the Maxwell-Boltzmann system (statistics), the probability of each state (elementary event) is $n^{-r}$.

The Bose-Einstein system is defined as a system of $r$ indistinguishable particles, each of which independently of the others can be in one of $n$ cells. Since the particles are indistinguishable, each state of this system is given by «filling numbers” $r_{1}, r_{2}, \ldots, r_{n}$, where $r_{j}$ is the number of particles in the cell No. $j$. If in this case all states of the system are considered equiprobable, then we speak of Bose-Einstein statistics. Thus, the probability of each state (elementary event) in the Bose-Einstein system is $\left(C_{n+r-1}^{n-1}\right)^{-1}$ (see formula (18)).

Note that if in the Bose-Einstein system we additionally require that no cells remain empty in each state of the system (clearly, this should be $r \geq n$ ), then the number of possible states of the system will be reduced to $C_{r-1}^{n-1}$ (this we proved above, in part b) of the last lemma).

The Fermi-Dirac system is defined as a Bose-Einstein system, which in addition to the Pauli exclusion principle requires that no more than one particle is in each cell.
Since in this case the particles are also indistinguishable, the state of the system is characterized by the numbers $r_{1}, r_{2}, \ldots, r_{n}$, where $r_{j}=0$ or $r_{j}=1$ (because in each cell there can be no more than one particle), $j=1,2, \ldots, n$, and mandatory $r \leq n$.

You can specify the system status by specifying the filled cells. The latter can be selected by $C_{n}^{r}$ ways ( $r$ cells can be selected from $n$ cells for $r$ particles in $C_{n}^{r}$ ways), the Fermi-Dirac system has the same number of states. If all states are equiprobable, then we speak of Fermi-Dirac statistics. Thus, the probability of each state (elementary event) in Fermi-Dirac statistics is $\left(C_{n}^{r}\right)^{-1}, r \leq n$.

Example 8. $r$ distinguishable (for example, numbered) particles are arranged in $n$ cells according to the Maxwell-Boltzmann system.
Find the probabilities of the following events:
a) Exactly $k(0 \leq k \leq r)$ particles fell in a certain cell (say, in cell No. 1).
b) Exactly $k(0 \leq k \leq r)$ particles fell in some cell.

## 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|Tasks for independent work

1. Winners of sports competitions are encouraged by the following awards: a certificate of honor (event $A$ ); a cash prize (event $B$ ), a medal (event $C$ ).
What do the following events mean:
a) $A | B$;
b) $A B C$;
c) $A B \backslash C$ ?
2. Prove an equality of events $\overline{\bar{A} \bar{B} \cup A}$ and $B \backslash A$.
3. Two people play chess. Let $A$ – the first player won, $B$ – the second player won. Describe the following events (below and everywhere further, $\Delta$ is the symmetric difference operation: for any events $C$ and $D: C \Delta D=(C \backslash D)+(D \backslash C))$ :
a) $A \Delta \bar{B}$
b) $\bar{A} \Delta B$;
c) $\overline{A \Delta B} ;$ d) $\bar{B} / A$
e) $\bar{A} / B$.
4. Prove the following equality:
a) $A /(A / B)=A B$;
b) $\overline{A B}=A \cap B$;
c) $\overline{\bar{A} \cup \bar{B}}=A B$
d) $A \bigcup B=A B \bigcup(A \Delta B)$;
e) $\overline{A \triangle B}=A B \cup \bar{A} \bar{B}$
f) $A \triangle B=(\overline{A B}) \Delta(\overline{\overline{A B}})$;
g) $\overline{\bigcup_{i=1}^{n} A_{i}}=\bigcap_{i=1}^{n} \overline{A_{i}}$,
h) $\bigcap_{i=1}^{n} A_{i}=\bigcup_{i=1}^{n} \bar{A}_{i}$.
5. Of the many married couples at random one is chosen. The event $A={$ the husband is more than 30 years old}, $B={$ the husband is older than the wife $}, C={$ the wife is more than 30 years old .
a) Clarify the meaning of events: $A B C, A \backslash A B, A \bar{B} C$;
b) Show that $A \bar{C} \subseteq B$.
6. Let $A$ and $B$ be some events. Prove that:
a) $A \cup B=A B \Delta(A \Delta B)$;
b) $A \backslash B=A \Delta(A B) ;$ c) $(A \cup \bar{B}) \Delta(\bar{A} \cup B)=A \Delta B$
7. Prove that
$$(A \cup B)(A \cup \bar{B}) \cup(\bar{A} \cup B)(\bar{A} \cup \bar{B})$$
and
$$(A \cup B)(\bar{A} \cup \bar{B}) \cup(A \cup \bar{B})(\bar{A} \cup B)$$
are certain events, and
$$(A \cup B)(A \cup \bar{B}) \cap(\bar{A} \cup B)(\bar{A} \cup \bar{B})$$
is an impossible event.
8. $A_{1}, A_{2}, \ldots, A_{N}$ are any events. Prove that:
$$\bigcup_{n=l k=n}^{N} \bigcap_{k}^{N}=\bigcap_{n=1 k=n}^{N} \bigcup_{k}^{N}=A_{N}$$
9. Express the following events through events $A_{1}, A_{2}, A_{3}$ :
a) Only an event $A_{1}$ occurs;
b) $A_{1}$ and $A_{2}$ occur, but $A_{3}$ doesn’t occur;
c) All three events occur;
d) At least one event occurs;
e) At least two events occur; f) Only one event occurs;
g) Only two events occur;
h) No events occurred;
i) At most two events occur.

## 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|Distribution of balls in boxes

\Omega=\left{\left(i_{1}, i_{2}, \ldots, i_{r}\right): i_{j} \in \Omega_{0}, j=1,2, \ldots , r\right}=\underbrace{\Omega_{0} \times \Omega_{0} \times \ldots \times \Omega_{0}}_{r}\Omega=\left{\left(i_{1}, i_{2}, \ldots, i_{r}\right): i_{j} \in \Omega_{0}, j=1,2, \ldots , r\right}=\underbrace{\Omega_{0} \times \Omega_{0} \times \ldots \times \Omega_{0}}_{r}

b) 向目标射击时，子弹对应于球，目标对应于盒子。
c) 在宇宙射线实验中，落入盖革计数器的粒子扮演球的角色，而计数器本身就是盒子。
d) 电梯离开（上升）r人和停在n楼层。然后，根据他们退出的楼层，将人分成组的分布对应于r球进n盒子。
e) 投掷实验r骰子对应的分布r球进n=6盒子。如果实验包括投掷r对称硬币，然后n=2.

## 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|Statistics of Maxwell-Boltzmann

Maxwell-Boltzmann 系统被描述为一个系统r可区分的（不同的）粒子，每个粒子都可以在n细胞（状态），无论剩余粒子位于何处。在这样的系统中，可能有nr不同的安排r粒子进入n细胞。如果这样做时，所有这样的安排（系统的状态）都被认为是同样可能的，那么我们说的是 MaxwellBoltzmann 统计。因此，在麦克斯韦-玻尔兹曼系统（统计学）中，每个状态（基本事件）的概率为n−r.

a) 完全正确ķ(0≤ķ≤r)粒子落入某个牢房（例如，在 1 号牢房）。
b) 完全正确ķ(0≤ķ≤r)颗粒落入一些细胞。

## 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|Tasks for independent work

1. 以下奖项鼓励体育比赛的获胜者：荣誉证书（活动一种); 现金奖励（活动乙), 奖牌 (事件C）。
以下事件是什么意思：
a)一种|乙;
b)一种乙C;
C）一种乙∖C ?
2. 证明事件相等一种¯乙¯∪一种¯和乙∖一种.
3. 两个人下棋。让一种– 第一个玩家获胜，乙– 第二个玩家赢了。描述以下事件（下面和任何地方，Δ是对称差分运算：对于任何事件C和D:CΔD=(C∖D)+(D∖C))：
一）一种Δ乙¯
b)一种¯Δ乙;
C）一种Δ乙¯;d)乙¯/一种
和）一种¯/乙.
4. 证明以下等式：
a)一种/(一种/乙)=一种乙;
b)一种乙¯=一种∩乙;
C）一种¯∪乙¯¯=一种乙
d)一种⋃乙=一种乙⋃(一种Δ乙);
和）一种△乙¯=一种乙∪一种¯乙¯
F）一种△乙=(一种乙¯)Δ(一种乙¯¯);
G）⋃一世=1n一种一世¯=⋂一世=1n一种一世¯,
h)⋂一世=1n一种一世=⋃一世=1n一种¯一世.
5. 从众多已婚夫妇中随机选择一对。事件一种=$吨H和H在sb一种nd一世s米这r和吨H一种n30是和一种rs这ld,乙={吨H和H在sb一种nd一世s这ld和r吨H一种n吨H和在一世F和}, C={吨H和在一世F和一世s米这r和吨H一种n30是和一种rs这ld.一种)Cl一种r一世F是吨H和米和一种n一世nG这F和在和n吨s:ABC, A \反斜杠 AB, A \bar{B} C;b)小号H这在吨H一种吨A \bar{C} \subseteq B$。
6. 让一种和乙是一些事件。证明：
a)一种∪乙=一种乙Δ(一种Δ乙);
b)一种∖乙=一种Δ(一种乙);C）(一种∪乙¯)Δ(一种¯∪乙)=一种Δ乙
7. 证明
(一种∪乙)(一种∪乙¯)∪(一种¯∪乙)(一种¯∪乙¯)

(一种∪乙)(一种¯∪乙¯)∪(一种∪乙¯)(一种¯∪乙)
是某些事件，并且
(一种∪乙)(一种∪乙¯)∩(一种¯∪乙)(一种¯∪乙¯)
是不可能的事件。
8. 一种1,一种2,…,一种ñ是任何事件。证明：
⋃n=lķ=nñ⋂ķñ=⋂n=1ķ=nñ⋃ķñ=一种ñ
9. 通过事件表达以下事件一种1,一种2,一种3:
a) 只有一个事件一种1发生；
b)一种1和一种2发生，但是一种3不会发生；
c) 所有三个事件都发生；
d) 至少发生一个事件；
e) 至少发生两个事件；f) 只发生一个事件；
g) 只发生两个事件；
h) 没有事件发生；
i) 最多发生两个事件。

## 广义线性模型代考

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MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。