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In this module we introduce the concept of self information for
an outcome of a stochastic variable.
Bergen, Norway is a rainy city. If the locals are "lucky" there
is "only" 200 rainy days in a particular year.Let the random variable Z take the two values: "Rain", "No rain".
Assuming 200 rainy days a year, we get
and
.We state that
carries more information than
,
the reason is that the inhabitans of Bergenexpect rain, so whenever it's not raining they are (more) surprised.
An intuitive definition of an information measure should be largerwhen the probability is small.
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The information content in a statement about the temperature
and new lottery millionaires in Verdal,Norway on a given saturdayshould be
the sum of the information on temperature
on the particular saturday in Verdal and the information of the numberof new lucky lottery winners, (under the assumption that these observations
are independent). Let I denote the information of an event, then
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An intuitive and meaningful measure of self information in an event should have
the following properties:
- The more
uncertain you, in advance, are about
the outcome, the
more new information you
get by observing the actual outcome, or equivalently an eventwith low probability,
, has high self information
.
should be a monotonically decreasing function of
.
- Oberserving an event with certain outcome, i.e
, should give zero information. The event
is then said to have zero self information. Since
is monotonically decreasing for
this implies that the self information can never
be less than zero, the observer can never lose informationby observing an outcome.
- If we receive independent messages, the information should
accumulate. This means that the measure must be additive.
It can be shown that there only exists one function satisfying the above conditions.
In the above equation the logarithm base can be chosen arbitrary.
Usually
is chosen so that the denomination is
information bit .
The choice
is made to adapt to a digital "world", that is to
facilitate electronic storage and transmission.
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Source:
OpenStax, Information and signal theory. OpenStax CNX. Aug 03, 2006 Download for free at http://legacy.cnx.org/content/col10211/1.19
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