Geometric Probabilities and Probability Distribution Laws

On a segment $AB$ of length $l$, two points $L$ and $M$ are placed at random. What is the probability that point $L$ turns out to be closer to point $A$ than point $M$? All positions of the points $L$ and $M$ on the segment $AB$ are equally possible.

On a segment $AB$ of length $l$, two points $L$ and $M$ are placed at random. What is the probability that point $L$ is closer to point $M$ than to point $A$? All positions of the points $L$ and $M$ are equally possible.

On a segment $AB$ of length $l$, a point $L$ is placed at random. What is the probability that the distance from this point to the midpoint of the segment is less than the distance from this point to one of the endpoints, and what is the probability that the distance to the midpoint is less than the distance to a specific endpoint? All positions of the point $L$ on the segment $AB$ are equally possible.

On a segment $AB$ of length $l$, two points $L$ and $M$ are placed at random. Compute the m.e. of the length of the segment $LM$ and the value of its standard deviation, given that all positions of the points $L$ and $M$ on the segment $AB$ are equally possible.

On a segment $AB$ of length $l$, two points $L$ and $M$ are placed at random. Assuming that all positions of the points $L$ and $M$ are equally possible, construct the cumulative and differential distribution curves of the length of the segment $LM$.

What is the probability that a triangle can be formed from three segments taken at random? The length of each of the segments does not exceed $a$, and all its values are equally possible.

A rod of length $l$ is broken into two parts, after which the larger part is broken in two again. Compute the probability that a triangle can be formed from the resulting parts.

Assuming that both roots of the equation

have equal probability of taking any value in the intervals $-1 \leqslant \alpha \leqslant 1$; $-1 \leqslant \beta \leqslant 1$, determine the probability densities $f(p)$ and $\varphi (q)$ of the coefficients $p$, $q$.

What is the probability that all the roots of the cubic equation

written at random, turn out to be real? Here it is assumed that all values of the coefficients $p$ and $q$ are equally possible and do not exceed the value $M$ in absolute value.

Two people agreed to meet at a certain place. As to the time, they agreed that each of them would be at the meeting place between 1 o'clock and 2 o'clock, and whoever arrives first would wait for the other for a quarter of an hour. Compute the probability that the meeting will take place.

Two people, meeting at an appointed place, have equal probability of arriving at any moment of the time interval $T$. Determine the probability $\mathbb {P}\left(P\right)(t_{0})$ that one of the two will have to wait for the other a time $t \leqslant t_{0}$, and find $\mathbb {E}\left[t\right]$ and $\mathbb {E}\left[t^{2}\right]$.

A rough diamond broke into $K$ pieces. Assuming that the value of an uncut diamond is proportional to the square of its weight, determine by what factor its value decreased, by computing the m.e. of the value of the fragments. In solving the problem, regard all possible divisions of the volume into $K$ parts as equally probable.

On the circumference of a circle, two points $A$ and $B$ are placed at random. Regarding all positions of the points on the circumference as equally possible, compute the probability density of the ratio of the distance between them to the diameter of the circle being equal to a given value $x$. Construct the probability distribution curve.

A material point describes an elliptical trajectory under the action of a central force. At a certain moment of time, the distance of the moving particle from the attracting center is measured. Compute the m.e. of the distance of the moving point from the center. All moments of measurement are equally probable. The major axis of the orbit is $a$ and the eccentricity is $e$.

Prove that, under any probability law, the m.e. of the square of the distance between two points $M_{1}(x_{1},y_{1})$ and $M_{2}(x_{2},y_{2})$ of a given region is no greater than twice the m.e. of the square of the distance from any point of the region to the origin, to which it is equal if $\mathbb {E}\left[x_{1}\right] = \mathbb {E}\left[y_{1}\right] = 0$.

Two points $x$ and $y$ lie on two opposite sides of a rectangle, whose sides equal $a$ and $b$; assuming a uniform probability distribution for each point, determine the probability $\mathbb {P}\left(P\right)(z_{1})$ that the distance between them $z < z_{1}$, as well as $\mathbb {E}\left[z\right]$ and $\mathbb {E}\left[z^{2}\right]$.

Solve a problem differing from the previous one (problem 6273 — points $x$ and $y$ on the sides of a rectangle with sides $a$ and $b$, uniform distribution) in that the points $x$ and $y$ lie on adjacent sides of the rectangle.

Determine the m.e. of the square of the distance between two points lying on any of the four sides of a rectangle, if all positions of the points are equally possible. Verify that it depends only on the perimeter $2p$.

On a segment $AB$ of length $l$, 2 segments of equal length $a$ are placed at random. Compute the m.e. of the length covered by these segments.

On the segment $(a,b)$, $n$ points $x_{i}$ are scattered at random (so that the probability law of $x_{i}$ on $(a,b)$ is uniform). It is required to determine the probability distribution law $p_{m}(x_{m})$ of the point $x_{m}$ occupying the $m$-th place from the end $b$.

Keeping the conditions of the previous problem (problem 6279 — $n$ points scattered at random on $(a,b)$), compute the m.e. of $x_{m}$ and its variance $\sigma^{2}(x_{m})$.

Show, keeping the conditions of problem 6279 ($n$ points scattered at random on $(a,b)$), that if $a$ and $b$ are given constant quantities, and the numbers $m$ and $n$ increase without bound in such a way that $\mathbb {E}\left[x_{m}\right] = 0$, then the probability of the inequality $\frac{\beta }{\sqrt{n}} < x_{m} < \frac{\alpha }{\sqrt{n}}$ has as its limit $\frac{1}{\sqrt{2\pi ab}}\int_{\alpha }^{\beta } e^{-z^{2}/2ab}\, dz$, where $\alpha$ and $\beta$ are arbitrary finite numbers.

Find, under the conditions of problem 6279 ($n$ points scattered at random on $(a,b)$), the equation of the correlation surface between $x_{k}$ and $x_{m}$, and show that both regression lines are straight.

Find, under the conditions of problem 6279 ($n$ points scattered at random on $(a,b)$), the probability distribution density $p(z)$ of the distance $z = x_{m}-x_{k}$, setting $k-m=h$.

Find, under the conditions of problem 6279 ($n$ points scattered at random on $(a,b)$), the limiting probability law of $n(x_{m}-x_{n})$ as $x \to \infty$, if $h=k-m$ remains finite.

Compute, under the conditions of problem 6279 ($n$ points scattered at random on $(a,b)$), the probability $u(\delta )$ that the smallest of the distances $x_{i}-x_{i+1}$ ($x_{0}=a$, $x_{n+1}=b$) is greater than a given value $\delta \leqslant \frac{b-a}{n+1}$. Consider the case $n \to \infty$.

Compute the probabilities $u_{1}(\delta )$ and $u_{2}(\delta )$ that exactly one and exactly two of the distances $x_{i}-x_{i+1}$ (under the conditions of problem 6279) are less than $\delta$.

A quantity $a$, which at $t=0$ equals zero, has probability $p\Delta t$ of increasing by one unit and probability $1-p\Delta t$ of remaining unchanged. Determine the limit of the probability that $a=k$ after the elapse of time $t$, letting $\Delta t \to 0$.

Event $A$ occurs $m$ times during a time period $T_{0}$, with the probability of its occurrence being the same at every moment. Determine the probability that it occurred $k$ times during a given interval $T$, letting $T_{0} \to \infty$ and $\frac{m}{T_{0}} \to p$.

The occurrence of event $A$ in $n$ observations was accompanied by the occurrence of event $B$. Determine the probability $\mathbb {P}\left(P_{m}\right)$ that, in $m$ further observations, the occurrence of event $A$ will be accompanied by the occurrence of event $B$. All assumptions regarding the original value of the probability of event $A$ occurring are equally possible.

A prize marksman, in $n$ shots, hit the center field of the target. How many observations must one have in order to expect, with probability not less than 0.999, a hit in the center on a new shot? Before information about the results of the shooting was received, all assumptions about the probability of the marksman hitting the center of the target were regarded as equally possible.

The occurrence of event $A$ in $n$ observations was accompanied by the occurrence of event $B$. How many observations must one have in order to expect, with probability not less than 0.999, the occurrence of event $B$ at the next ten occurrences of event $A$? The original assumptions about the value of the probability of event $A$ occurring are equally possible.

A segment $AB$ consists of two inhomogeneous parts. A series of samples, taken at random at various points of this segment, gave $m$ samples belonging to one part, and $n$ samples belonging to the other. What ratio between the parts appears most probable?

A disk has holes whose sizes and arrangement are difficult to measure. $N$ shot pellets are thrown at the disk, of which $M$ fell through the holes. Regarding a pellet as a point, and assuming that before the experiment all assumptions about the ratio between the area of the disk and the areas of the holes are equally possible, establish the most probable ratio between the areas of the holes and the area of the disk.

Compute the characteristic function $\theta (t) = \int_{-\infty }^{\infty } e^{itx}f(x)\, dx$ for the continuous probability distribution curve

Compute the characteristic function $\theta (t) = \int_{-\infty }^{\infty } e^{itx}f(x)\, dx$ for the continuous probability distribution curve $f(x) = 0$ for $\left|x\right| > 2$; $f(x) = \frac{1}{2}\left(1-\frac{\left|x\right|}{2}\right)$ for $\left|x\right| \leqslant 2$.

Compute the characteristic function $\theta (t) = \int_{-\infty }^{\infty } e^{itx}\, dF(x)$ for the probability distribution corresponding to the cumulative distribution law $F(x)$: $F(x)=0$ for $x<a$; $F(x)=p$ for $a \leqslant x < b$; $F(x)=1$ for $x > b$.

Compute the characteristic function $\theta (t) = \int_{-\infty }^{\infty } e^{itx}\, dF(x)$ for the probability distribution corresponding to the cumulative distribution law $F(x)$: $F(x)=0$ for $x<a$, $F(x)=1-e^{-2k(x-a)}$ for $x \geqslant a$.

Compute the characteristic function $\theta (t) = \int_{-\infty }^{\infty } e^{itx}\, dF(x)$ for the probability distribution corresponding to the cumulative distribution law $F(x)$: $F(x)=0$ for $x<-a$, $F(x)=\frac{x+a}{2a}$ for $-a \leqslant x < a$, $F(x)=1$ for $x \geqslant a$.

Compute the characteristic function and the moments of the distribution of problem 6270 (the ratio of the distance between two points on a circle to the diameter of the circle).

Assuming all the functions introduced in the previous problem (problem 6300) to be continuous and differentiable, establish the relations between them and the function $p(x,b)$, expressing the probability density for a passenger who boarded at point $x$ to get off at point $b > x$.

Determine the traffic-density function $\varphi (x)$, given that the probability $\varphi '_{1}(x)$ of boarding the tram at point $x$ is proportional to $x(L-x)^{2}$, and the probability $p(x,y)$ is proportional to $(y-x)^{h}$, where $h \geqslant 0$. Consider the cases $h=1$ and $h=0$.

Determine $p(x,y)$ and $\varphi (z)$, if the probability of a route from $x$ to $y$, given by $\frac{\partial^{2} F(x,y)}{\partial x \partial y}$, is proportional to $(y-x)^{h}$, where $h \geqslant 0$. Determine, for each value of $h$, the ratio $\mathbb {P}\left(P_{h}\right)$ of the maximal traffic density to the mean density; investigate how $\mathbb {P}\left(P_{h}\right)$ varies as a function of $h$.

Determine $\varphi (z)$, if $p(x,y)$ is proportional to $\sin \frac{\pi y}{L}$, and $\varphi_{1}(z) = \sin \frac{\pi z}{2L}$ (i.e. the boarding probability $\varphi '(z) = \frac{\pi }{2L}\cos \frac{\pi z}{2L}$).

Determine the boarding probability $\varphi '_{1}(z)$ at point $z$ and the traffic density $\varphi (z)$, if the probability of the route $(x,y)$ equals 0 for $y-x < a$ and is proportional to $x(L-y)$ for $y-x \geqslant a$ (consider the case $a = \frac{L}{2}$).

Find the limiting probability distribution law of $y_{n}$ as $n \to \infty$, if

where $x_{i}$ are random variables, independent of each other, for which $\mathbb {E}\left[x_{i}\right] = 0$, $\mathbb {E}\left[x_{i}^{2}\right] = 1$, $\mathbb {E}\left[\left|x_{i}\right|^{3}\right] < c^{3}$, with $p$, $k$, $c$ given numbers (the limit theorem of Lyapunov--Markov is assumed known).

Find the m.e. of $x_{i}^{2}$ and the limiting probability law for $S_{n} = \sum_{i=0}^{n} x_{i}$ as $n \to \infty$, if $x_{i+1} = \left(1-\frac{1}{A}\right)x_{i} + \alpha_{i+1}$, where $\alpha_{i}$ are independent random variables, with $\mathbb {E}\left[\alpha_{i}\right] = 0$, $\mathbb {E}\left[\alpha_{i}^{2}\right] = b^{2}$, assuming that the ratio $\frac{\mathbb {E}\left[\left|\alpha_{i}\right|^{3}\right]}{b^{3}} < \delta$ is bounded, $A > 1$, $x_{0}=0$ and $\lim_{n\to \infty }\frac{A}{n} = 0$.

Keeping the conditions of the previous problem (problem 6307), find the limit of the correlation coefficient $R_{\lambda }$ between $S_{n}$ and $S_{n_{1}}$, where $n_{1} = \lambda^{2}n$; $\lambda^{2} < 1$.

Solve the two preceding problems (problems 6307 and 6308) under the condition that $\lim \frac{A}{n} = a$ is different from 0. Consider the case where $a \to \infty$, with $b^{2} = \frac{K}{nA^{2}}$, where $K$ is a finite number.

Find the limiting probability density $p(y)$ of the distribution of $y_{n}$ as $n \to \infty$, if

assuming that $\alpha_{i}$ are independent random variables that take only the values $+1$, $-1$ with probabilities equal to $\frac{1}{2}$.

Find the limiting probability law $\mathbb {P}\left(P\right)(y)$ for $y_{n}$, if

where $\alpha_{i}$ takes the values $qa$ and $-pa$ ($a>0$) with respective probabilities $p$ and $q$.

Compute the correlation coefficient $R_{n+1}$ between $\Delta y_{n} = y_{n+1}-y_{n}$ and $\Delta y_{n+1} = y_{n+2}-y_{n+1}$, if the quantities $y_{n}$ are independent and $\mathbb {E}\left[y_{n}\right] = a$, setting $\mathbb {E}\left[y_{n}^{2}\right] = \sigma_{n}^{2}$; verify that $R_{i} < 0$; show that as $n \to \infty$, $R_{n} \to -\frac{1}{2}$, if $\lim \sigma_{n} = \sigma$ for all $i$.

For a certain production process to take place, $k$ independent factors $A_{1}, A_{2}, \ldots , A_{k}$ are required, which finish their work in times $x_{1}, x_{2}, \ldots , x_{k}$, so that the whole process finishes after a time $X$ equal to the largest of the values $x_{i}$, $i=1,2,\ldots ,k$. It is required to determine the m.e. of the total time loss $\sum_{k}$ of all the factors, if the cumulative probability laws $\mathbb {P}\left(P_{i}\right)(x_{i})$ of each factor are known. Consider the case where $\mathbb {P}\left(P_{i}\right)(x) = \mathbb {P}\left(P\right)(x)$ does not depend on $i$.

Solve the previous problem (problem 6313 — $k$ independent factors, the process completion time equal to the largest of the $x_{i}$) in the case where $\mathbb {P}\left(P\right)(x) = 0$ for $x < a$; $\mathbb {P}\left(P\right)(x) = \sin^{2}\frac{\pi }{2b}(x-a)$ for $a < x < a+b$; $\mathbb {P}\left(P\right)(x) = 1$ for $x > a+b$.