4. Geometric definition of probability
The event A = {a point falls in the region g}. Then
,
Chapter II. Basic formulas
5. Conditional probability
Definition
5.1 Let
A,B
be
events and
.
The probability of the event A
with
additional condition: the event B
occurred,
is called the conditional probability of the event A.
Notation:
.
.
Theorem 5.1(Theorem of multiplication of probabilities)
.
Definition 5.2 We say that an event A is independent of an event B if P(A/B) = P(A). Otherwise A dependent of B.
Theorem 5.2 If A is independent of B, then B is independent of A. This means that the property of independence is mutual.
.
Theorem
5.3
If events A
and
B
are
independent, then pairs of events
are independent.
Theorem 5.4 If A and B are independent, then
.
6. Formula of total probability. Bayes’ formulas
Let events
satisfy the following conditions:
a).
;
b).
.
Definition 6.1 If satisfy conditions a),b) then we say that is a full group of events.
Theorem
6.1(Formula
of total probability)
Let
be a full group of events, B
be
an event such that
,
i.e.
and
are mutually exclusive if
.
Then
(1)
Theorem 6.2 (Bayes’ formulas) Let be a full group of events and an event B satisfies all conditions of theorem 6.1. Then
.
Chapter III. Bernoulli scheme
7. Bernoulli’ formula
Definition 7.1 Events are pairwise independent if
.
Definition
7.2 Events
jointly independent, if for any
and for any
and
are
independent.
Theorem
7.1 Events
jointly independent
for
any
.
Definition 7.3 A random experiment is called a Bernoulli scheme, or a sequence of n independent identical tests with two outcomes if
These n tests are jointly independent, i.e. any n outcomes of these n tests are jointly independent.
As a result of every test we have exactly two outcomes (one outcome we call success and denote A and another we call failure and denote
).The probability
is the same in every test, so
.
Theorem
7.2
.
8. The most probable number of successes in n tests of Bernoulli
If
is a point of maxima, then
So
if
is an integer, then
has two values:
and
.
If
is not integer, then
.
Definition 8.1 is called the most probable number of successes in n tests of Bernoulli.
9. Approximated Poisson formula. Moivre-Laplace theorems
Theorem
9.1 If
p
is small and
.
Then
.
Theorem 9.2 (Local Moivre-Laplace theorem) If a probability p of an event A is a constant in each test and 0 < p <1, then we have the approximated formula for :
,
where
,
is a tabular function.
Theorem 9.3 (Integral Moivre-Laplace theorem) Let p satisfies all conditions of theorem 9.2. Then we have the following approximated formula:
,
where
,
is the Laplace function (tabular function),
.
Chapter IV. Random variables
10. Definition of random variable. Function of distribution
Definition
10.1
Let
be
a random variable and
where R
is
the set of all real numbers. Then
is called the function of distribution of the random variable .
Definition 10.2 is called a random variable if
It’s values depend of chance.
It has a function of distribution.
Example
10.2 Let
A
be
an event of Bernoulli scheme, P(A)
= p and
n
the number of tests. If
is the number of occurring of A,
then
and
is the random variable. We have
.
Properties
of
1.
.
2.
.
3. is not decreasing function.
4.
.
5. is continuous from the right.
Definition 10.3 Any connections between values of a random variable and probabilities corresponding to these values are called a law of distribution of a random value.