Bernoulli's Theorem. Chebyshev's Inequalities

Applying the Chebyshev--Markov lemma, indicate a lower bound for the probability $\mathbb {P}\left(P\right)$ of the inequality

for $n$ independent trials, if in each trial the probability equals $\frac{1}{2}$, taking $n = 10\, 000$; $n = 100\, 000$.

Applying the Chebyshev--Markov lemma, determine how large $n$ must be so that the probability $\mathbb {P}\left(P\right)$ of inequality (*) from problem 6222 ($\left|\frac{m}{n} - \frac{1}{2}\right| < \frac{1}{100}$, for $n$ independent trials with probability $\frac{1}{2}$ in each trial) is greater than 0.99?

Applying the Chebyshev--Markov lemma, for $n = 10\, 000$, what upper bound for $\left|\frac{m}{n} - \frac{1}{2}\right|$ can be indicated with probability not less than 0.99?

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, applying the Chebyshev--Markov lemma, that in 10000 throws he will lose at least 200 rubles?

Assuming that in $n$ successive independent trials the probabilities of event $A$ occurring are respectively $p_{i} = \left(\frac{1}{2}\right)^{i}$, determine a lower bound for the probability that the number of its occurrences $m \leqslant 1$.

Show that, whatever the random variable $x$ and the given positive number $\varepsilon$ may be, the probability of the inequality

is less than $e^{-t^{2}}$, where $I = \mathbb {E}\left[e^{\varepsilon x}\right]$.

Letting $m$ be the number of occurrences of event $A$ in a series of independent trials with respective probabilities $p_{i}$, derive from (A) — inequality $\varepsilon x > t^{2} + \lg I$ from problem 6227 — that the probability of the inequality

is less than $e^{-t^{2}}$ (if $\varepsilon \leqslant 1$, $t^{2} \leqslant \sum_{n=1}^{n} p_{i}q_{i}$).

Show, by the same method as in problem 6228, that the probability of the inequality

is less than $e^{-t^{2}}$ ($t^{2} \leqslant \sum_{i=1}^{n} p_{i}q_{i}$).

Using inequalities (B) and (C) from problems 6228 and 6229, solve problems 6223 and 6225.

Show that if $p = \frac{1}{2}$, then the probability of the inequality

is less than $e^{-t^{2}}$. Apply this to the solution of problem 6224.

$n$ independent trials are performed, where the probability $p_{i}$ of event $A$ occurring in the $i$-th trial equals $\left(\frac{3}{4}\right)^{i}$; determine an upper bound for the probability $p$ that the number $m$ of occurrences of $A$ will be more than 10.

The quantity $x_{k}$ takes three values: $+k^{\beta }$, $-k^{\beta }$ and 0 with probabilities equal respectively to $\frac{1}{2k^{\alpha }}$, $\frac{1}{2k^{\alpha }}$ and $1-\frac{1}{k^{\alpha }}$. Prove that the law of large numbers applies to the sum of independent quantities $\sum_{k=1}^{n} x_{k}$, if $\alpha > 1$.

Generalize the result of the previous problem (problem 6233), by showing that the law of large numbers is applicable to the sum $\sum_{k=1}^{n} x_{k}$ of independent random variables $x_{k}$, defined as follows. The random variable $x_{k}$ takes a series of values with probabilities given in the table:

The series $p_{1}+p_{2}+\ldots +p_{k}+\ldots$ is convergent.