The Theorems of Laplace and Lyapunov--Markov

Using Laplace's formula, indicate how many times $n$ a die must be thrown so that the probability of the inequality

is not less than the probability of the opposite inequality, where $m$ is the number of occurrences of 6.

A player wins 7 rubles when 6 comes up and pays 1 ruble in other cases. It is required to indicate, with probability $2\Phi (4) \approx 0{,}999936$, between what limits his winnings $R$ will lie after 8000 throws of the die.

The probability of an event occurring in each trial of a series of independent trials is the same and equals $\frac{3}{4}$. With what probability can one expect that, in 10000 trials, the ratio of the number $m$ of occurrences of the event to the number $n$ of trials performed differs from the probability $\frac{3}{4}$ by less than 0.0001, less than 0.001, and less than 0.01, respectively?

Buffon, performing an experiment to check the law of large numbers, tossed a coin 4040 times. Heads appeared 2048 times. Compute the probability with which one can expect that, upon repeating Buffon's experiment, the deviation of the frequency of heads from the probability will not exceed the deviation that occurred in Buffon's experiment.

For an event having probability of occurrence $p = 0{,}2$, a series of independent trials will be performed. Indicate what deviation of the frequency of the event's occurrence from the probability can be expected with probability $\mathbb {P}\left(P\right) = 0{,}9128$ in 5000 trials.

Experiments were performed to establish the resistance of various steel alloys to corrosion.

Experiments performed for a certain alloy showed that this alloy does not succumb to corrosion for 875 days on average. This determination is accompanied by a standard deviation equal to 85 days.

Using the probability integral, indicate how many sheets manufactured from this alloy must be subjected to testing so that, with probability 0.90, one can expect the deviation from the mean to be no more than $5\%$.

From an urn containing black and white balls in the ratio $1:3$, $n$ draws of a ball are made. At each draw the color of the ball is recorded, after which it is returned to the urn. What number $n$ should be chosen so that the deviation of the frequency of occurrence of a white ball from its probability can be expected to be no more than 0.001, with probability 0.9966?

How many times should a coin be tossed so that, with probability 0.6, one can expect the deviation of the frequency of heads from the probability 0.5 to turn out less than 0.01?

Determine the equation of the envelope of all normal curves $y = \frac{1}{\sigma \sqrt{2\pi }}e^{-x^{2}/2\sigma^{2}}$.

In counting the number of primes that fall, in the eighth, ninth, and tenth millions, into each of 2000 groups of 500 consecutive numbers, the following data were found: no group turned out to have fewer than 17 primes or more than 46. In the top row of the accompanying table are placed all the numbers $x$ ($17 \leqslant x \leqslant 46$) actually observed; in the following three rows, in each column under the corresponding value of $x$, is indicated the number of groups $N_{1}(x)$, $N_{2}(x)$, $N_{3}(x)$ containing $x$ primes in the eighth, ninth, and tenth million, respectively.

Compute the mean numbers $M_{1}$, $M_{2}$, $M_{3}$ of primes per group and their variances $\sigma_{1}^{2}$, $\sigma_{2}^{2}$, $\sigma_{3}^{2}$ for each million. Show that if, in each million, the primes were arranged at random, then the variances would be considerably larger.

Show that if the mean values $M_{1}$, $M_{2}$, $M_{3}$ indicated in the previous example (problem 6244) and the corresponding standard deviations $\sigma_{1}^{2}$, $\sigma_{2}^{2}$, $\sigma_{3}^{2}$ had been obtained as a result of mechanical (random) sampling, then the difference between $M_{1}$ and $M_{3}$ could not be regarded as random; on the contrary, the difference between $M_{1}$ and $M_{2}$ could well be random.

Construct, according to the Poisson scheme, a collection of groups of 500 trials each, where the probability $p_{i}$ of a prime occurring would equal 0 or $p$ in the $i$-th trial, so that the mean probability equals $\mathbb {P}\left(P\right) = \frac{1}{16} = 0{,}0625$ for a given variance $\sigma^{2} = 18\frac{11}{48} = \frac{875}{48}$. Show, using elementary considerations of divisibility of numbers, that in a natural sequence of 500 numbers there cannot be more than 119 primes.

In two localities the age distribution of workers is characterized by the table:

Determine whether the difference in the ratio of the number of workers over 35 to the total number of workers in a given locality appears greater than what could be expected under random sampling.

Steel wire used in the manufacture of cables withstands, on average, a tensile force of $6720 \text{ kg/cm}^{2}$. The standard deviation of individual samples from this mean has the value $220 \text{ kg/cm}^{2}$. A series of successive observations showed that, with respect to the tensile forces withstood by individual wire samples, a normal distribution law holds.

From each batch of wire arriving at the plant, 20 specimens were subjected to testing. The test results are given in the table below.

Value of the tensile force withstood by the wire per 1 cm2

Is it possible, on the basis of these tests, to assert that both batches of wire are of the required quality and that the test results should be explained by chance of sampling?

Using the data of the previous problem (problem 6248 — tensile tests of wire from two batches), what could be said about the quality of the wire arriving at the plant if the test log had not indicated which batch each specimen belonged to?

The acceptance of batches of wire (see problem 6249) takes place under conditions where it turns out that neither the mean value of the tensile force nor its standard deviation for the wire arriving at the plant had previously been determined. Can one say in this case that the quality of the incoming batches of wire is different, or should the test result be attributed to chance of sampling?

Tensile tests performed for the same alloy, prepared at five different plants, gave the following result:

The tensile force and its standard deviation are expressed in kilograms per square centimeter.

Should the difference in means be explained by the random selection of specimens, or by a difference in the quality of the alloys, i.e. by a deviation in the production of one of the plants from the standard grade?

For a shaft of radius $r$ to be able to rotate in a circular hole of a bearing of radius $R$, it is necessary that free space remain between the shaft and the hole. The value of the difference between $R$ and $r$ must, however, remain within known limits

In the serial manufacture of shafts and bearings, both parts are prepared separately, so that upon assembly it may turn out that the parts do not fit one another. Assuming that in a well-adjusted production process the deviations of the quantities $r$ and $R$ from the mean values $r_{0}$ and $R_{0}$ follow a normal distribution law, compute what percentage of the items will have to be discarded during assembly on the conveyor.

The distribution of values of the quantity $x$ has standard deviation $\sigma$. An individual measurement of the values of $x$ is accompanied by a mean square error $\sigma '$. To measure $x$ one must use specimens, the manufacture of which costs $a$ rubles apiece. The cost of carrying out one measurement is $a'$ rubles. How many individual specimens should be manufactured, and how many measurements should be made on each, so that the mean value of the quantity $x$, established as a result of this series of observations, is accompanied by a mean square error not exceeding $\alpha$, and the cost of carrying out the experiment is as small as possible?

Prove that if $x_{k}$ can take, with probabilities equal to $\frac{1}{2}$, two values $\pm k^{\lambda }$, where $\lambda \geqslant -\frac{1}{2}$, then the Lyapunov--Markov theorem is applicable to the sum $S_{n} = \sum_{k=1}^{n} x_{k}$.

Consider a series of a very large number $n$ of trials, broken up into groups of 3 each so that, in the first two trials of each group, the probability of event $A$ occurring equals $\frac{1}{2}$, while the outcome of the third trial is predetermined by the results of the preceding two trials in such a way that the number of occurrences of $A$ in each group must be even (0 or 2). Show that the number $m$ of occurrences of event $A$ in the whole series of $n$ trials obeys the same limiting Laplace--Gauss law as if all the trials were independent (as in the Bernoulli scheme).

The quantity $x_{k}$ can take the values $\pm k^{\lambda }$ with probabilities $\frac{1}{2k^{\lambda }}$, or the value 0 with probability $1-\frac{1}{k^{\lambda }}$, where $0 < \lambda < 1$. It is required to prove that the Lyapunov--Markov theorem is applicable to the sum $\sum_{k=1}^{n} x_{k}$ of the independent quantities $x_{k}$.