数学专业英语(12)分析解析
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Mathematical English 12: Probability Theory
and Mathematical Statistics
Mathematical
English
Dr. Xiaomin Zhang
Email: zhangxiaomin@
Dr. Xiaomin Zhang: Mathematics
Department, School of Science, Ningbo University
1
Mathematical English 12:
Probability Theory and Mathematical Statistics
§
2.12 Probability Theory and
Mathematical Statistics
TEXT A Special terminology peculiar to
probability theory
In
discussions
involving
probability,
one
often
sees
phrases
from
everyday language such as
“
two events are equally
likely,
”
“
an event is
impossible,
”
or
“
an
event
is
certain
to
occur.
”
Expressions
of
this
sort
have
intuitive appeal and it is both pleasant and
helpful to be able to
employ such
colorful language in mathematical discussions.
Before we
can
do
so,
however,
it
is
necessary
to
explain
the
meaning
of
this
language in terms of the fundamental
concepts of our theory.
Because
of the
way probability is
used
in practice,
it is
convenient
to
imagine that each probability space
(
S,
B
,
P
) is associated with a real or
conceptual experiment. The universal
set
S
can then be thought of
as
the collection of all conceivable
outcomes of the experiment, as in the
Dr. Xiaomin Zhang: Mathematics
Department, School of Science, Ningbo University
2
Mathematical English 12:
Probability Theory and Mathematical Statistics
example
of
coin
tossing
discussed
in
the
foregoing
section.
Each
element of
S
is
called an
outcome
or a
sample
and the subsets of
S
that
occur in
the Boolean algebra
B
are
called
events
. The reasons
for this
terminology will become more
apparent when we treat some examples.
Assume
we
have
a
probability
space
(
S,
B
,
P
)
associated
with
an
experiment.
Let
A
be
an
event,
and
suppose
the
experiment
is
performed and that its outcome is
x.
(In other words, let
x
be a point of
S.
) This outcome
x
may or may not belong to
the set
A
. If it does, we
say that the event
A
has occurred. Otherwise, we say that
the event
A
has not
occurred, in which case
x
A'
,
so the complementary event
A'
has occurred.
An event
A
is called
impossible
if
A=
, because in
this
case no outcome of the experiment
can be an element of
A
. The
event
A
is
said
to
be
certain
if
A=S
,
because
then
every
outcome
is
automatically an element of
A.
Each
event
A
has
a
probability
P(A)
assigned
to
it
by
the
probability
Dr. Xiaomin Zhang: Mathematics
Department, School of Science, Ningbo University
3
Mathematical English 12:
Probability Theory and Mathematical Statistics
function
P
.
(The
actual
value
of
P(A)
or
the
manner
in
which
P(A)
is
assigned is not concern
us at present.) The number
P(A)
is also called
the
probability that an
outcome of the experiment is one of the elements
of A.
We also say that
P(A)
is the
probability that
the event A
occurs
when the experiment is
performed.
The impossible event
must be assigned
probability zero because
P
is
finitely additive measure. However,
there may be events with probability
zero
that
are
not
impossible.
In
other
words,
some
of
the
nonempty
subsets
of
S
may
be
assigned
probability
zero.
The
certain
event
S
must be
assigned probability 1 by the very definition of
probability, but
there may be other
subsets as well that are assigned probability 1.
In
example 1 of Section 6.8 there are
nonempty subsets with
probability
zero and proper subsets of
S
that have probability 1.
Two events
A
and
B
are said to be
equally likely
if
P(A)=P(B)
. The event
A is called
more
likely
than
B
if
P(A)>P(B),
and
at
least as likely as B if
Dr. Xiaomin
Zhang: Mathematics Department, School of Science,
Ningbo University
4
Mathematical English 12: Probability
Theory and Mathematical Statistics
P(A)
P(B).
Table
2-12-1
provides
a
glossary
or
further
everyday
language that is often used in
probability discussions. The letters
A
and
B
represent
events,
and
x
represents
an
outcome
of
an
experiment
associated with the sample space
S
. Each entry in the left-
hand column
is a statement about the
events
A
and
B
, and the corresponding
entry in
the right-hand column defines
the statement in terms of set theory.
Dr. Xiaomin Zhang: Mathematics
Department, School of Science, Ningbo University
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Mathematical English 12:
Probability Theory and Mathematical Statistics
Notations
probability function
here
the value of probability function P at point A
is
the
probability
that
the
event
A
occurs.
Generally,
The
probability
function
P(x)
(also
called
the
probability
density
function
or
density
function) of a
continuous distribution is defined as the
derivative of the
(cumulative)
distribution function D(x),
so
A probability function satisfies
Dr. Xiaomin
Zhang: Mathematics Department, School of Science,
Ningbo University
6
Mathematical English 12: Probability
Theory and Mathematical Statistics
and
is constrained by the normalization condition,
Special cases
are
To find the probability
function in a set of transformed variables, find
the
Jacobian. For example, If u=u(x),
then
Dr.
Xiaomin Zhang: Mathematics Department, School of
Science, Ningbo University
7
Mathematical English 12: Probability
Theory and Mathematical Statistics
so
Similarly, if
u=u(x, y) and v=v(x, y), then
Given n probability
functions P
1
(x),
P
2
(x), ...,
P
n
(x), the sum distribution
X+Y+
…
+Z has
probability function
where
(x) is a
delta function. Similarly, the probability
function for the
distribution of
XY
…
Z is given by
Dr. Xiaomin Zhang: Mathematics
Department, School of Science, Ningbo University
8
Mathematical English 12:
Probability Theory and Mathematical Statistics
The difference distribution
X-Y has probability function
and the ratio distribution
X/Y has probability function
Dr. Xiaomin Zhang: Mathematics
Department, School of Science, Ningbo University
9
Mathematical English 12:
Probability Theory and Mathematical Statistics
TEXT
B
two
basic
statistics
concepts
—
population
and
sample
In the
preceding sections, we cited a few examples of
situations where
evaluation
of
factual
information
is
essential
for
acquiring
new
knowledge.
Although
these
examples
are
drawn
from
widely
differing
fields and only sketchy descriptions of
the scope and objectives of the
studies
are
provided,
a
few
common
characteristics
are
readily
discernible.
First,
in
order
to
acquire
new
knowledge,
relevant
date
must
be
collected. Second, some amount of
variability in the data is unavoidable
even though observations are made under
the same or closely similar
conditions.
For
instance,
the
treatment
for
an
allergy
may
provide
long-lasting
relief
for
some
individuals
whereas
it
may
bring
only
transient relief or
even none at all to others. Likewise, it is
unrealistic to
Dr. Xiaomin Zhang:
Mathematics Department, School of Science, Ningbo
University
10