On groups and series of successes in Bernoulli sequences of random variables

S. M. Ananjevskii, V. B. Nevzorov

Saint Petersburg State University

Abstract: The paper considers sequences of independent identically distributed random variables $X_1,X_2,\ldots,$ taking values $ 0, 1$ with probabilities
$$ \mathbf{P} \{X_n=0\}=1-p, \quad\mathbf{P} \{X_n=1\}=p, $$
where $0<p<1. $ The distributions and their properties for series and groups of successes are studied. The research started in the authors' previous papers is continued.

Key words and phrases: Bernoulli scheme, binomial distribution, geometric distribution, negative binomial distribution, mathematical expectation, generating functions.

UDC: 519.2

Received: 09.09.2025