### 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|Geometric probabilities

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|Geometric probabilities

Let $\Omega={\omega}$ be a bounded subset of an $n$-dimensional Euclidean space $R^{n}$. We will assume that for $\Omega$ the concept of «volume» makes sense (for $n=1$ – length, for $n=2$ – area, for $n=3$ – usual volume, etc.) We denote by $\beta=\beta(\Omega)$ the system of subsets of $\Omega$ (events), which have «volumes») and for any event $A \in \beta(\Omega)$ we will determine its probability by the relation
$$P(A)=\frac{\operatorname{mes}(A)}{\operatorname{mes}(\Omega)}$$
where mes $(A)$ is the «volumen) of the event (the set) $A$.
The definition of probability by the formula (1) is called a geometric definition of probability.

The constructed model can be considered as a model of an experiment consisting of random throwing of a point into the domain $\Omega$ (Here and in the following we will understand an expressions of the type «The point is randomly thrown into the area $\Omega$ » or “The random point is uniformly distributed in the domain $\Omega »$ as «The point dropped at random to the area $\Omega$ can reach any point of the area $\Omega$, and the probability of this point falling into some part $A$ of the area $\Omega$ is proportional to the “volume») of this part and does not depend on the form and location of this part in $\Omega »$ ).

## 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|A random point

1. A random point is placed on a segment of length $l$ (say, a segment $[0, l]$ ), as a result, the segment is divided into two parts.

Find the probability that the length of a larger segment does not exceed $4 /(5 l)$ (event $A$ ).

Solution. Denote by $x$ the length of one of the segments, then the length of the second segment is equal to $l-x$ (Fig, 1 ).
Fig. 1
Them the sample space is
$$\Omega={x: 0 \leq x \leq l}=[0, l]$$
and the desired event

$$A=\left{x \in \Omega: \max (x, l-x) \leq \frac{4}{5} l\right}=\left[\frac{1}{5} l, \frac{4}{5} l\right]$$
Therefore, we have by formula (1)
$$P(A)=\frac{\operatorname{mes}(A)}{\operatorname{mes}(\Omega)}=\frac{\frac{3}{5} l}{l}=\frac{3}{5} .$$

1. At the random moment of time $x$ a signal of length $\Delta$ appears on the time segment $[0, T]$. The receiver is switched on at a random time point $y \in[0, T]$ for a time $t$. Find the probability of detecting the signal by the receiver.
Solution. The sample space is the domain
$$\Omega={(x, y): \quad 0 \leq x, y \leq T}=[0, T] \times[0, T] .$$
If first a signal appears, and the receiver is connected later, i.e. if $x \leq y$, then the signal is detected only when $y-x \leq \Delta$.

Similarly, if $y \leq x$, then the signal can be detected only in the case if $y \geq x-t$ Thus, the event we need
$$A={(x, y) \in \Omega: y-x \leq \Delta, y \geq x \text {, t.e. } x-y \leq t, x \geq y}$$
is the area that is shaded in the Fig. $2 .$

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

1. Three points are placed at random into the semi-straight line $[0, \infty)$.
Find the probability that we can make a triangle from the segments formed from the point zero $(\leftrightarrow 0))$ to the given three points.
1. Two points are placed at random into the segment of length of $l$.
Find the probability that a triangle can be made from the three formed segments.
2. Three points, one after another, are put at random on the segment of a line. Find the probability of hitting a third point between the first two points.
3. A random point $X$ is placed on a segment $A B$ of length $a$, then a random point $Y$ is placed on a segment of length $b$.

Assuming that the points $A, B, C$ are on the line in this order, find the probability of forming a triangle from the segments $A X, B Y, X Y$.

1. A random point is thrown into the sphere of radius $R$.
Find the probability that the distance from this point to the center of the sphere does not exceed $r$.
2. A random point is placed in the square.
Find the probability that the distance from this point to the vertices of the square exceeds half of the length of the side of the square.
3. A random point $A$ is placed in the square with the side $a$.
Find the probability that the distance from $A$ to the nearest side of the square does not exceed the distance from $A$ to the nearest diagonal of the square.
4. The point $X$ is randomly placed on a semicircumference $C=\left{(x, y): x^{2}+y^{2}=R^{2}, y \geq 0\right}$. Find the probabilities of the following events:
a) the abscissa of the point $X$ lies on the segment $[-r, r]$;
b) the ordinate of the point lies on the segment $[r, R]$.
5. The plane is marked with parallel straight lines at the same distance $a$ from each other. The coin (circle) of radius $r\left(r<\frac{a}{2}\right)$ is randomly thrown to the plane.
Find the probability that the coin does not intersect any straight line.
6. The Bertrand Paradox. Two points are randomly chosen in a circumference of radius $r$. They are connected by a chord.

Find the probability that the length of the chord will exceed $\sqrt{3} r$ (that is, the length of the side of an equilateral triangle inscribed in the circle).

1. Contimuation. The point is randomly chosen in a circumference of radius $r$; a diameter is drawn through it. A random point (the middle of the chord that is perpendicular to the diameter) is taken on the diameter.
Find the probability that the length of the obtained chord will surpass $\sqrt{3} r$.
2. Continuation. The point is placed at random inside a circle of radius $r$. This point is the middle of the chord that is perpendicular to the diameter passing through it.
Find the probability that the length of the obtained chord will surpass $\sqrt{3} r$.
3. Two points are placed at random into segments $[-a, a],[-b, b], a>0, b>0, p$ and $q$ are their coordinates (respectively).

Find the probability that the roots of the quadratic equation $x^{2}+p x+q=0$ are real numbers.

1. The segment of length of $a_{1}+a_{2}$ is divided into two parts of the length $a_{1}$ and $a_{2}$, respectively. The $n$ points are randomly placed on this segment.
Find the probability that exactly $m$ out of $n$ points will be placed on a part of the length $a_{1}$.
2. Continuation. The segment of length $a_{1}+a_{2}+\ldots+a_{s}$ is divided into $s$ parts of the length $a_{1}, a_{2}, \ldots, a_{s}$. The $n$ points are randomly placed on this segment.

Find the probability that $m_{1}, m_{2}, \ldots, m_{s}\left(m_{1}+m_{2}+\ldots+m_{s}=n\right)$ points will be placed on parts of lengths $a_{1}, a_{2}, \ldots, a_{s}$ (respectively).

## 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|A random point

1. 一个随机点放置在一段长度上l（比如说，一段[0,l])，因此，该段被分为两部分。

Ω=X:0≤X≤l=[0,l]

1. 在随机的时间X长度信号Δ出现在时间段上[0,吨]. 接收机在随机时间点开启是∈[0,吨]有一段时间吨. 求接收机检测到信号的概率。
解决方案。样本空间是域
Ω=(X,是):0≤X,是≤吨=[0,吨]×[0,吨].
如果首先出现信号，然后接收器连接，即如果X≤是，那么只有当信号被检测到是−X≤Δ.

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

1. 三个点随机放入半直线[0,∞).
找出我们可以从零点形成的线段组成三角形的概率(↔0))到给定的三点。
1. 将两个点随机放入长度为l.
找出可以从三个形成的线段组成三角形的概率。
2. 三个点一个接一个地随机放置在一条直线上。找出在前两点之间击中第三点的概率。
3. 一个随机点X放置在一个段上一种乙长度一种，然后是一个随机点是放置在一段长度上b.

1. 一个随机点被扔进半径球体R.
求这个点到球心的距离不超过的概率r.
2. 在正方形中放置一个随机点。
求该点到正方形顶点的距离超过正方形边长一半的概率。
3. 一个随机点一种被放置在有边的正方形中一种.
找到距离的概率一种到广场最近的一侧不超过距离一种到正方形最近的对角线。
4. 重点X随机放置在一个半圆周上C=\left{(x, y): x^{2}+y^{2}=R^{2}, y \geq 0\right}C=\left{(x, y): x^{2}+y^{2}=R^{2}, y \geq 0\right}. 求下列事件的概率：
a) 点的横坐标X位于段上[−r,r];
b) 点的纵坐标位于线段上[r,R].
5. 平面用等距的平行直线标出一种从彼此。半径的硬币（圆）r(r<一种2)被随机扔到飞机上。
求硬币不与任何直线相交的概率。
6. 伯特兰悖论。在半径的圆周上随机选择两个点r. 它们通过和弦连接。

1. 延续。该点是在半径的圆周中随机选择的r; 通过它绘制一个直径。在直径上取一个随机点（垂直于直径的弦的中点）。
求得到的和弦长度超过的概率3r.
2. 继续。该点随机放置在半径圆内r. 该点是垂直于通过它的直径的弦的中点。
求得到的和弦长度超过的概率3r.
3. 两个点被随机放置成段[−一种,一种],[−b,b],一种>0,b>0,p和q是它们的坐标（分别）。

1. 段的长度一种1+一种2长度分为两部分一种1和一种2， 分别。这n点随机放置在该段上。
找到确切的概率米在……之外n点将放置在长度的一部分上一种1.
2. 继续。长度段一种1+一种2+…+一种s分为s部分长度一种1,一种2,…,一种s. 这n点随机放置在该段上。

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