Mathematical Expectations. The Method of Finite Differences and Generating Functions

In a lottery two items are raffled: a bicycle worth 2500 rubles and a watch worth 500 rubles. What is the mathematical expectation of the winnings for a person holding one ticket, if the total number of winning tickets is 1000?

From a box containing 4 white and 20 black balls, 5 balls are drawn and transferred to another box, in which 16 white balls had been placed; after this, 6 balls are transferred back from the second box to the first. What will be the mathematical expectations of the numbers of white balls $x_{1}$ and $x_{2}$ in both boxes after this?

In throwing 3 dice, a player wins 100 rubles if all the dice show 6, and receives 10 rubles if six comes up on only 2 of the dice. What sum must he pay for a ticket for the right to take part in the game so that the game be fair?

In throwing $n$ dice, determine the m.e. (mathematical expectation) of the sum of points on all the dice and the variance of this sum.

The probability of event $A$ occurring equals $p$. Compute the m.e. $x$ of the number of successive trials in which $A$ repeats without interruption, and the variance $B$ of this number.

For each run of $a > 0$ trials in which $A$ repeats without interruption, a player receives $y^{a}$ rubles; on the other hand, if $a = 0$, he pays 1 ruble. Determine the value of $y$ under the condition that the game be fair. Consider the example $p = \frac{1}{13}$.

Under the conditions of problem 6167 (from a vessel containing $m$ white and $n$ black balls, a ball is drawn and returned to the vessel, the drawing continuing until a white ball appears), find $\mathbb {E}\left[M\right]$.

From a vessel containing $m$ white and $n$ black balls, we draw balls in succession until a white ball appears. Determine the m.e. of the number of black balls drawn in this process.

From a vessel containing $m$ white and $n$ black balls, balls are drawn and returned, until a white ball appears. Compute the m.e. of the number of black balls drawn and its variance.

A vessel contains $a$ tickets numbered 1, $b$ tickets numbered 2, , and $k$ tickets numbered $m$. From the vessel $n$ tickets are drawn. Compute the m.e. of the number of numbers that did not come out.

A vessel contains $a$ tickets numbered 1, $b$ tickets numbered 2, $c$ tickets numbered 3, and so on, $k$ tickets numbered $m$, where $a+b+\ldots +k \geqslant m$. Determine the m.e. of the number of coincidences of the number of a drawn ticket with the number of the draw, when all the balls are drawn.

Prove that if $\mathbb {E}\left[x^{2}\right] = \mathbb {E}\left[x^{3}\right] = \mathbb {E}\left[x^{4}\right] = a$, then the quantity $x$ cannot take values other than 0 and 1.

It is known that $x \leqslant A$ and $\mathbb {E}\left[x\right] = 0$; show that the probability $\mathbb {P}\left(P\right)$ of the inequality $x < -At^{2}$ is less than $\frac{1}{1+t^{2}}$.

Prove the identity

(where $A \geqslant a$ are positive integers), by computing in two different ways the m.e. of the number $m$ of black balls separating 2 neighboring white balls in a series consisting of $A+1$ balls, among which there are $a$ white ones.

Show that the variance $\sigma^{2}$ of the quantity $x$ cannot exceed the square of half the difference between its extreme values. Indicate the conditions under which this upper bound is attained.

The quantity $x$ can take all positive integer values with probabilities decreasing in geometric progression. Determine the common ratio $q$ of the progression so that $\mathbb {E}\left[x\right] = 10$, and under this condition compute the probability $\mathbb {P}\left(P_{10}\right)$ that $x \leqslant 10$, and the probability $\mathbb {P}\left(P_{100}\right)$ that $x \leqslant 100$.

The quantity $x$ can take the value 0 with probability $a_{0}$ and any positive integer value $n > 0$ with probabilities $p_{n}$ decreasing in geometric progression ($p_{n} = B^{n}$). It is required to determine the indicated probabilities, given that $\mathbb {E}\left[x\right] = a$, $\mathbb {E}\left[x^{2}\right] = b$, and to establish the conditions that $a$ and $b$ must satisfy for the problem to be possible. Check whether the values $a=4$, $b=25$ are admissible.

The quantity $x$ takes nonnegative integer values $n \geqslant 0$ with probabilities $a_{n} = A\frac{k^{n}}{n!}$. Determine $A$ and $k$, given that $\mathbb {E}\left[x\right] = a$, and find the most probable value $\mu$ of the number $n$.

Find the necessary and sufficient conditions that must be imposed on $a = \mathbb {E}\left[x\right]$, $b = \mathbb {E}\left[x^{2}\right]$, $c = \mathbb {E}\left[x^{3}\right]$, in order that the quantity $x$ take only nonnegative integer values $n \geqslant 0$ with probabilities $a_{n} = \frac{Ak^{n}+Bl^{n}}{n!}$, where $A \geqslant 0$, $B \geqslant 0$, $k \geqslant 0$, $l \geqslant 0$. Consider the numerical example: $a = \frac{7}{4}$, $b = \frac{13}{2}$, $c = \frac{13}{4}$; determine the most probable value of $n$.

Solve the previous problem (problem 6206) under the condition that $k$ may be negative.

Example. $a = 1{,}85$; $b = 5{,}7$; $c = 20{,}95$.

Prove that $\left(\mathbb {E}\left[x^{p}\right]\right)^{q} \geqslant \left(\mathbb {E}\left[x^{q}\right]\right)^{p}$, if $x \geqslant 0$ and $p > q > 0$.

Prove that, whatever the quantity $x$ may be, the moments, i.e. the m.e.'s of its successive powers $a_{i} = \mathbb {E}\left[x^{i}\right]$, satisfy the inequalities

A number $x \geqslant 0$ can take any integer value with probability

Compute $a_{h}$, given that $\mathbb {E}\left[x\right] = A$; show that $\mathbb {E}\left[x^{2}\right]$ has no meaning. Setting $A=3$, compute the probability $\mathbb {P}\left(P_{10}\right)$ that $x < 10$.

It is known that

represents the probability that a number $x \geqslant 0$ takes the integer value $h$. Determine $a_{h}$ under the condition that the standard deviation $\sigma$ of the number $x$, as well as its m.e. $A$, be integers.

Two boxes with balls are given; the first has $N$ balls, of which $M$ are white, the second has $N_{1}$ balls, of which $M_{1}$ are white. Simultaneously one ball from one box is exchanged for a ball from the other, and conversely. This mixing operation is performed in succession $l$ times. Determine the m.e. of the number of white balls in each of the boxes after this. Consider the case $l \to \infty$.

Solve the previous problem (problem 6213) under the assumption that $N_{1} = \infty$, i.e. under the condition that a ball drawn from the first box is always replaced by a black one. Consider the case $l \to \infty$, with $\frac{l}{N} = a$ remaining finite.

Compute the variance $B_{l}$ of the number of white balls after $l$ draws under the conditions of the previous problem (problem 6214: from the first box with $N$ balls, of which $M$ are white, a drawn ball is always replaced by a black one, with $l \to \infty$, $\frac{l}{N}=a$ finite).

The probability of an event occurring changes from trial to trial. Construct the generating function of the probabilities $\mathbb {P}\left(P'_{m,n}\right)$ of the event occurring $m$ or fewer times in $n$ independent trials.

In a certain game a player has equal probability $p=\frac{1}{5}$ of winning nothing, or of winning or losing 20 or 40 kopecks. Determine the probability that after 12 rounds he will lose at least 3 rubles, and the variance $B$ of his winnings.

If a player, in throwing a die, wins 4 rubles when 6 comes up and pays 1 ruble in all other cases, what is the probability that in 10000 throws he will lose at least 200 rubles?

Assuming that, in compiling a 7-digit table (mantissas) of logarithms, all digits are equally probable, what is the probability $\mathbb {P}\left(R_{s}\right)$ that the sum $S$ of the digits of the mantissa turns out to be less than 20? What are $\mathbb {E}\left[S\right]$ and $\mathbb {E}\left[S^{2}\right]$?

A number $x$ can take any integer values $m$ with probability $(1-q)q^{m}$; adding $h$ values of this number, determine the probability $\mathbb {P}\left(P_{a}\right)$ that the resulting sum $y$ equals $a$. Find the most probable value $S$ of this sum; consider the limiting case where $h \to \infty$, with $hq = t$ remaining finite.

A number $x_{n}$, when $n$ increases by 1, receives an increment of $+1$ or $-1$, both values being equally possible if $x_{n}=0$; if $x_{n} \geqslant 0$, then the probability of a decrease in $\left|x_{n}\right|$ equals $A \geqslant \frac{1}{2}$ (the probability of an increase in $\left|x_{n}\right|$ equals $B = 1-A \leqslant \frac{1}{2}$). Setting $x_{0}=0$, find the probability $\mathbb {P}\left(P_{n,h}\right)$ that $\left|x_{n}\right|=h$ for any $n$ and $h$; consider the case $n \to \infty$. Compute $\mathbb {E}\left[\left|x_{n}\right|\right]$. Consider the case $A=B=\frac{1}{2}$.