Questions and Answers of Theory Of Probability

Prove that the function defined by the equations\[ f(t)=f(-t), \quad f(t+2 a)=f(t), \quad f(t)=\frac{a-t}{a} \text { for } 0 \leqslant t \leqslant a \]is a characteristic function.The characteristic
What meaning do the following equations have:(a) \(A B C=A\);(b) \(A+B+C=A\) ?
Simplify the expressions(a) \((A+B)(B+C)\);(b) \((A+B)(A+\bar{B})\);(c) \((A+B)(A+\bar{B})(\bar{A}+B)\).
Prove the equations:(a) \(\overline{\bar{A} \bar{B}}=A+B\);(b) \(\overline{\bar{A}+\bar{B}}=A B\);(c) \(\overline{A_{1}+A_{2}+\cdots+A_{n}}=\ddot{A}_{1} \bar{A}_{2} \ldots \bar{A}_{n}\)(d)
A four-volume work is on a shelf in random order. What is the probabilty that the volumes stand in the proper order from left to right or from right to left?
The numbers \(1,2,3,4,5\) are written on five cards. Three cards are drawn in succession and at random from the deck; the resulting digits are written from left to right. What is the probability that
There are \(M\) defective items in a lot consisting of \(N\) items. From this lot we select \(n(n
A quality control inspector examines items in a lot consisting of \(m\) items of first grade and \(n\) second-grade items. An inspection of the first \(b\) items chosen at random from the lot showed
Using probabilistic arguments, prove the identity \((A>a)\) :\[ 1+\frac{A-a}{A-1}+\frac{(A-a)(A-a-1)}{(A-1)(A-2)}+\ldots+\frac{(A-a) \ldots 2 \cdot 1}{(A-1) \ldots(a+1) a}=\frac{A}{a} \]An urn has
Draw one ba 1 after another in succession from a box containing \(m\) white balls and \(n\) black balls \((m>n)\). What is the probability that there will come a time when the number of selected
A person wrote letters to \(n\) addressees, one letter in each envelope, and then, at random, wrote one of the \(n\) addresses on each envelope. What is the probability that at least one of the
An urn has \(n\) tickets with numbers from 1 to \(n\). The tickets are drawn at random, one at a time (without replacement). What is the probability that at least in one selection the number of the
From an urn containing \(n\) white balls and \(n\) black ones select at random an even number of balls (all the different ways of drawing an even number of balls are considered equally probable,
The paradox of de Méré. What is more probable: to get one ace with four dice, or to get one double ace in 24 throws of two dice?
Three points are thrown at random on a segment \((0, a)\). Find the probability that a triangle can be constructed out of line-segments equal to distances from point 0 to the points of fall.
A rod of length \(l\) is broken at two randomly chosen points. What is the probability that the pieces can be used to build a triangle?
A point is dropped at random onto line-segment \(A B\) of length \(a\). Another point is dropped at random on a line-segment \(B C\) of length \(b\). What is the probability that a triangle can be
A total of \(N\) points are dropped at random and independently of one another into a sphere of radius \(R\).(a) What is the probability that the distance from the centre to the nearest point will be
The events \(A_{1}, A_{2}, \ldots, A_{n}\) are independent; \(P\left(A_{k}\right)=p_{k}\). Find the probability of:(a) the occurrence of at least one of these events;(b) the nonoccurrence of all
Prove that if events \(A\) and \(B\) are mutually exclusive, \(P(A)>0\) and \(P(B)>0\), then events \(A\) and \(B\) are dependent.
Let \(A_{1}, A_{2}, \ldots, A_{n}\) be random events. Prove the formula\[ \begin{aligned} P\left\{\sum_{k=1}^{n} A_{k}\right\}=\sum_{i=1}^{n} P\left(A_{j}\right)- & \sum_{1 \leqslant i
The probability that a molecule that collided with another molecule at time \(t=0\) and that did not experience any other collisions up to time \(t\) will experience a collision during the time
Assuming that in the multiplication of bacteria by fission (division into two bacteria) the probability of a bacterium dividing during a time interval \(\Delta t\) is equal to \(a \Delta t+o(\Delta
A workman operates 12 machines of the same type. The probability that one machine will require his attention during a time interval of duration \(\tau\) is \(1 / 3\). What is the probability that:(a)
A certain family has 10 children. Considering the probability of birth of a boy and a girl equal to \(1 / 2\), find the probability that in this family(a) there are 5 boys and 5 girls;(b) the number
In a gathering of 4 persons, the birthdays of three come in one month and that of the fourth in one of the remaining eleven months. Considering the probability of birth of each person in each month
In 14,400 tosses of a coin, heads fell 7,428 times. How probable is such a large or larger deviation of the number of heads from \(n p\) if the coin is symmetric (that is, the probability of throwing
A total of \(n\) devices, each with a power consumption of \(a\) kilowatts, are connected to an electric network. At a given time each is consuming power with a probability \(p\). Find the
An educational institution has a student body of 730 . The probability that the birthday of a randomly selected student will fall on a definite day of the year is \(1 / 365\) for each of the 365
It is known that the probability of producing a drill bit of extra-high brittleness (defective) is 0.02 . The bits are packed in boxes of a hundred each. What is the probability that(a) a box will
An insurance company has issued policies to 10,000 persons of the same age and the same social group. The probability of death during the year for each person is 0.006 . On January 1 each insured
Prove the following theorem: if \(P\) and \(P^{\prime}\) are the probabilities of the most probable number of occurrences of an event \(A\) in \(n\) and \(n+1\) independent trials (in each of the
In the Bernoulli scheme, \(p=1 / 2\). Prove that(a)\[\begin{gathered} \frac{1}{2 \sqrt{n}} \leqslant P_{2 n}(n) \leqslant \frac{1}{\sqrt{2 n+1}} \\ \lim _{n \rightarrow \infty} \frac{P_{2 n}(n+t
Prove that for \(n p q \geqslant 25\)\[P_{n}(m)=\frac{1}{\sqrt{2 n p q}} e^{-\frac{z^{2}}{2}}\left[1+\frac{(q-p)\left(z^{3}-3 z\right)}{6 \sqrt{n p q}}\right]+\Delta\]where\[z=\frac{m-n p}{\sqrt{n p
A total of \(n\) independent trials have been performed. The probability of the occurrence of event \(A\) in the \(i\) th trial is \(p_{i} ; P_{n}(m)\) is the probability of the \(m\)-fold occurrence
Prove that for \(x>0\) the function \(\int_{x}^{\infty} e^{-\frac{z^{z}}{2}} d z\) satisfies the inequalities\[\frac{x}{1+x^{2}} e^{-\frac{1}{2} x^{2}} \leqslant \int_{x}^{\infty} e^{-\frac{1}{2}
A certain mathematician always carries two boxes of matches with him. Whenever he wants a match, he selects one of the boxes at random. Find the probability that when the mathematician draws an empty
A total of \(n\) machines are connected to an electric transmission line. The probability that a machine consuming power at time \(t\) will cease to consume up to time \(t+\Delta t\) is equal to
One workman operates \(n\) automatic machines of the same type. If at time \(t\) a machine is operating, then the probability that it will require attention prior to time \(t+\Delta t\) is equal to
The transition probabilities are given by the matrix\[ \pi_{1}=\left\{\begin{array}{ccc} \frac{1}{2} & \frac{1}{3} & \frac{1}{6} \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{6} \\ \frac{1}{2} &
An electron may reside in one of a countable set of orbits depending on its energy. Transition from the \(i\) th orbit to the \(j\) th orbit takes place in one second with a probability \(c_{i}
The transition probabilities are given by the matrix\[ \pi_{1}=\left\{\begin{array}{ccc} 0 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & 0 & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & 0
Prove that if \(F(x)\) is a distribution function, then for any \(h eq 0\) the functions\[ \Phi(x)=\frac{1}{h} \int_{x}^{x+h} F(x) d x, \quad \Psi(x)=\frac{1}{2 h} \int_{x-h}^{x+h} F(x) d x \]are
A random variable \(\xi\) has \(F(x)\) as its distribution function \((p(x)\) is the density function). Find the distribution function (density function) of the random variable:(a) \(\eta=a \xi+b,
From the point \((0, a)\) draw a straight line at an angle \(\varphi\) to the \(y\)-axis. Find the distribution function for the abscissa of the point of intersection of this line with the
A point is thrown at random on the circumference of a circle of radius \(R\) with centre at the coordinate origin [in other words, the polar angle of the point of impact is uniformly distributed in
A point is thrown at random on the segment of the ordinate axis between points \((0,0)\) and \((0, R)\) [that is, the ordinate of the point is uniformly distributed in the interval \((0, R)\) ].
The diameter of a circle is measured approximately. Considering that it is uniformly distributed in the interval \((a, b)\), find the distribution of the area of the circle.
The density function of a random variable \(\xi\) is given by the equation\[ p(x)=\frac{a}{e^{-x}+e^{x}} \]Find:(a) the constant \(a\);(b) the probability that in two independent observations
The distribution function of a random vector \((\xi, \eta)\) is of the form:(a) \(F(x, y)=F_{1}(x) F_{2}(y)+F_{3}(x)\);(b) \(F(x, y)=F_{1}(x) F_{2}(y)+F_{3}(x)+F_{4}(y)\).Can the functions
Two points are dropped at random on the interval \((0, a)\) [that is, their abscissas are uniformly distributed on the interval \((0, a)]\). Find the distribution function of the distance between
A total of \(n\) points are dropped on the interval \((0, a)\). Assuming that the points have been dispersed at random [that is, each of them is situated irrespective of the others and is distributed
A total of \(n\) independent trials are performed on a random variable \(\xi\) having a continuous distribution function, as a result of which the following vaiues of the variable \(\xi\) were
The distribution function of the random vector \(\left(\xi_{1}, \xi_{2}, \ldots, \xi_{n}\right)\) is \(F\left(x_{1}\right.\), \(\left.x_{2}, \ldots, x_{n}\right)\). As the result of a trial the
The random variable \(\xi\) has a continuous distribution function \(F(x)\). How is the random variable \(\eta=F(\xi)\) distributed?
The random variables \(\xi\) and \(\eta\) are independent; their density functions are defined by the equations\[ \begin{aligned} & p_{\xi}(x)=p_{\eta}(x)=0 \quad \text { for } x \leqslant 0 \\ &
Find the distribution function of the sum of the independent random variables \(\xi\) and \(\eta\), the first of which is uniformly distributed in the interval ( \(-h, h\) ), and the second has the
The density function of the random vector \((\xi, \eta, \zeta)\) is\[ ho(x, y, z)=\left\{\begin{array}{cc} \frac{6}{(1+x+y+z)^{4}} & \text { for } x>0, y>0, z>0 \\ 0 & \text { otherwise }
Find the distribution of the sum of the independent random variables \(\boldsymbol{\xi}_{1}\) and \(\xi_{2}\) if their distributions are given by the conditions:(a)
The density function of the independent random variables \(\xi\) and \(\eta\) is:(a) \(p_{\xi}(x)=p_{\eta}(x)=\left\{\begin{array}{cc}0 & \text { for } x0(a>0)\end{array}\right.\)(b)
Find the distribution function of the product of the independent factors \(\xi\) and \(\eta\) on the basis of their distribution functions \(F_{1}(x)\) and \(F_{2}(x)\).
The random variables \(\xi\) and \(\eta\) are independent and distributed as follows:(a) uniformly in the interval ( \(-a, a\) );(b) normally with parameters \(a=0, \sigma=1\).Find the distribution
The sides \(\xi\) and \(\eta\) of a triangle are independent random variables. Using their distribution functions \(F_{\xi}(x)\) and \(F_{\eta}(x)\) find the distribution function of the third side
Prove that if the variables \(\xi\) and \(\eta\) are independent and their density function is\[ p_{\xi}(x)=p_{\eta}(x)=\left\{\begin{array}{cc} 0 & \text { for } x0 \end{array}\right. \]then the
Prove that if the variables \(\xi\) and \(\eta\) are independent and normally distributed with parameters \(a_{1}=a_{2}=0, \sigma_{1}=\sigma_{2}=\sigma\), then the variables\[ \zeta=\xi^{2}+\eta^{2}
Prove that if the variables \(\xi\) and \(\eta\) are independent and distributed in accordance with the chi-square law with parameters \(m\) and \(n\), then the variables \(\delta=\frac{\xi}{\eta}\)
The random variables \(\xi_{1}, \xi_{2}, \ldots, \xi_{n}\) are independent and have one and the same density function\[ p(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{(x-a)^{2}}{2 \sigma^{2}}} \]Find
Prove that any distribution function possesses the following properties:\[ \begin{array}{ll} \lim _{x \rightarrow \infty} x \int_{x}^{\infty} \frac{1}{z} d F(z)=0, & \lim _{x \rightarrow+0} x
Two series of independent trials are performed with a random variable \(\xi\) which has a continuous distribution function \(F(x)\). As a result, \(\xi\) took on values arranged in the order of
The random variable \(\xi\) has a continuous distribution function \(F(x)\). As a result of \(n\) independent observations of \(\xi\), we have the following values \(x_{1}
The random variables \(\xi\) and \(\eta\) are independent and identically distributed with the density function\[ p_{\xi}(x)=p_{\eta}(x)=\frac{C}{1+x^{4}} \]Find the constant \(C\) and prove that
The random variables \(\xi\) and \(\eta\) are independent and their density functions are, respectively, given byProve that the variable \(\xi \eta\) is normally distributed. 1 P (x)= (x < 1) and P
Let \(\xi\) and \(\zeta\) be independent and let them have the density functions\[ p_{\xi}(x)=p_{\zeta}(x)= \begin{cases}0 & \text { for } x \leqslant 0 \\ \lambda e^{-\lambda x} & \text { for }
The random variables \(\xi\) and \(\eta\) are independent and are uniformly distributed on the interval \((-1,1)\). Compute the probability that the roots of the equation \(x^{2}+\xi x+\eta=0\) are
A random variable \(\xi\) takes on only integral nonnegative values with probabilities(a) \(\mathrm{P}(\xi=k)=\frac{a^{k}}{(1+a)^{k+1}}, a>0\) is a constant (this is the Pascal distribution).(b)
Let \(\mu\) be the number of occurrences of an event \(A\) in \(n\) independent trials, in each of which \(P(A)=p\). Find(a) \(M \mu^{3}\),(b) \(M \mu^{4}\),(c) \(M|\mu-n p|\)
The probability that event \(A\) will occur in the \(i\) th trial is \(p_{i}\). Let \(\mu\) be the number of occurrences of \(A\) in the first \(n\) independent trials. Find(a) \(M \mu\),(b)
Prove that, given the conditions of the preceding problem, \(D \mu\) reaches a maximum for the given value of \(a=\frac{1}{n} \sum_{1}^{n} p_{i}\) provided\[ p_{1}=p_{2}=\ldots=p_{n}=a \]Preceding
Let \(\mu\) be the number of occurrences of an event \(A\) in \(n\) independent trials, in each of which \(P(A)=p\). Also, let a variable \(\eta\) be 0 or 1 depending on whether \(\mu\) proves to be
The density function of a random variable \(\xi\) is\[ p(x)=\frac{1}{2 \alpha} e^{-\frac{|x-a|}{\alpha}} \](the Laplace distribution). Find \(\mathbf{M}_{\xi}\) and \(\mathbf{D \xi}\).
The density function of the absolute speed of a molecule is given by the Maxtell distribution\[ p(x)=\frac{4 x^{2}}{\alpha^{3} \sqrt{\pi}} e^{-\frac{x^{2}}{\alpha^{2}}} \text { for } x>0 \]and
The probability density that a molecule in Brownian motion will be at a distance \(x\) from a reflecting wall at time \(t\), if at time \(t_{0}\) it was at a distance \(x_{0}\), is given by the
Prove that for an arbitrary random variable \(\xi\), the possible values of which lie in the interval \((a, b)\), the following inequalities are valid:\[ a \leqslant M \xi \leqslant b \text { and }
Let \(x_{1}, x_{2}, \ldots, x_{k}\) be possible values of a random variable \(\xi\). Prove that as \(n \rightarrow \infty\)(a) \(\frac{M \xi^{n+1}}{M \xi^{n}} \rightarrow \max _{1 \leqslant j
Let \(F(x)\) be the distribution function of \(\xi\). Prove that if \(\mathbf{M} \xi\) exists, then\[ M \xi=\int_{0}^{\infty}[1-F(x)+F(-x)] d x \]and for the existence of \(\boldsymbol{M} \xi\) it
Two points are dropped at random on the line-segment \((0, l)\). Find the expectation, variance and the expectation of the nth power of the distance between them.
A random variable \(\xi\) is distributed according to the logarithmic normal law; i.e., for \(x>0\) the density function of \(\xi\) is\[ p(x)=\frac{1}{x \beta \sqrt{2 \pi}} e^{-\frac{1}{2
A random variable \(\xi\) is normally distributed. Find \(M|\xi-a|\) where \(a=M \xi\).
A box contains \(2^{n}\) tickets; the number \(i(i=0,1, \ldots, n)\) is written on \(C_{n}^{l}\) of them. A total of \(m\) tickets are drawn at random, \(s\) is the sum of the numbers written on
The random variables \(\xi_{1}, \xi_{2}, \ldots, \xi_{n+m}(n>m)\) are independent, identically distributed and have a finite variance. Find the correlation coefficient of the sums\[
The random variables \(\xi\) and \(\eta\) are independent and are normally distributed with the same parameters \(a\) and \(\sigma\). Find the correlation coefficient of the quantities \(\alpha
A random vector \((\xi, \eta)\) is normally distributed; \(\boldsymbol{M} \xi=a, \mathbf{M} \eta=b, \mathbf{D} \xi=\sigma_{1}^{2}\), \(D \eta=\sigma_{2}^{2}\), and \(R\) is the correlation
Let \(x_{1}\) and \(x_{2}\) be the results of two independent observations of a normally distributed variable \(\xi\). Prove that \(M \max \left(x_{1}, x_{2}\right)=a+\frac{\sigma}{\sqrt{\pi}}\)
A random vector \((\xi, \eta)\) is normally distributed, \(\mathbf{M} \xi=\mathbf{M} \eta=0, \mathbf{D} \xi=\mathbf{D} \eta=1\), \(\mathbf{M} \xi \eta=R\). Prove that\[ M \max (\xi,
The unevenness in length of cotton fibre is given by\[ \lambda=\frac{a^{\prime \prime}-a^{\prime}}{a} \]where \(a\) is the expectation of fibre length, \(a^{\prime \prime}\) is the expectation of
The random variables \(\xi_{1}, \xi_{2}, \ldots, \xi_{n}, \ldots\) are independent and uniformly distributed over \((0,1)\). Let \(v\) be a random variable equal to the \(k\) for which the sum\[
Let \(\xi\) be a random variable with density function\[ p_{\xi}(x)=\frac{1}{\pi} \frac{1}{1+x^{2}} \]Find \(M \min (|\xi|, 1)\).
Prove that if the random variable \(\xi\) is such that \(M e^{a *}\) exists \((a>0\) is a constant), then\[ \mathbf{P}\{\xi \geqslant \varepsilon\} \leqslant \frac{M
Let \(f(x)>0\) be a nondecreasing function. Prove that if \(\mathbf{M}(f(|\xi-M \xi|)\) exists, then\[ \mathbf{P}\{|\xi-\mathbf{M} \xi| \geqslant \varepsilon\} \leqslant \frac{\mathbf{M}
A sequence of independent and identically distributed random variables \(\left\{\xi_{i}\right\}\) is defined by the equalities(a) \(\mathbf{P}\left\{\xi_{n}=2^{k-\log k-2 \log \log

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