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BernoulliVersuch

BernoulliVersuch (Bernoulli trial) is a basic concept in probability theory describing a random experiment with two possible outcomes: success with probability p and failure with probability 1-p. The two outcomes are typically denoted as 1 for success and 0 for failure. Each trial is assumed independent of others and to maintain the same probability p across repetitions.

A Bernoulli random variable X takes the value 1 with probability p and 0 with probability 1-p.

When the experiment is repeated n times independently, the total number of successes S follows a binomial

A Bernoulli process is an infinite sequence of independent Bernoulli trials with the same p, used to

Related concepts include the Bernoulli distribution, which describes a single trial; the geometric distribution, which describes

Origin and usage: The concept is named after Jacob Bernoulli (1654–1705). BernoulliVersuche underpin the binomial distribution

Its
expected
value
is
E[X]
=
p
and
its
variance
is
Var(X)
=
p(1-p).
distribution
with
parameters
n
and
p.
The
probability
of
k
successes
is
P(S
=
k)
=
C(n,
k)
p^k
(1-p)^{n-k}.
model
events
that
occur
over
time.
the
number
of
trials
until
the
first
success;
and
the
negative
binomial
distribution,
which
generalizes
this
to
a
fixed
number
of
successes.
and
are
widely
used
in
statistics,
probability
modeling,
quality
control,
finance,
and
various
fields
of
science
to
model
binary
outcomes
and
to
derive
related
results.