Probability And Measure Theory

By Author: Pierce Brosnan
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Introduction to probability and measure theory:

Measure theory:
Measure theory is a branch of mathematical analysis which studies the notion of size of an abstract set and aggregates of a function.Measure theory broadens our vision about probability, random events, random variables and integration. Measure theory, to a large extent, is self contained, but preliminary concepts on set theory, functions and real analysis is essential.

Probability:
Probability provides mathematical models for random phenomena, that is, phenomena which under repeated observations yield different outcomes that cannot be predicted with certainty.

Probability and measure theory:

Probability:

Example 1:

Consider two identical coins.What is the probability that the coins match (both heads or both tails)?

Solutions:

If we toss two identical coins the set of possible outcomes is S = {HH,HT,TT} .
Stated in terms of the number of heads that appear , S ={2,1,0} . However, these simple events are not equally likely; there is only one way to get 2 heads and only one way to ...
... get 0 heads, but exactly 1 head can occur in two ways. If the coins were distinguishable, then
the sample space would be
R = {HH,HT,TH,TT}
and these events are equally likely with each having the probability 1/4 .

P(HH) = 1/4

P(TT) = 1/4

Let A = “the coins match”. Then A = {HH,TT} and P(A) = 1/2.

Example 2:

Consider two identical coins. What is the probability that at least one head turns up?

Solution:
If the coins were distinguishable, then the sample space would be
R = {HH,HT,TH,TT}
and these events are equally likely with each having the probability 1/4 .

Note that the compound event {HT,TH} in R corresponds to the simple event HT (or exactly 1 head) in S. Since the coins don’t know whether or not they are identical, we will assume that they are and then go on to determine the probability assignment for S:

P(HT) = 1/2

Let B = “at least 1 head”. Then B = {HH,HT,TH} and P(B) = 3/4.

Probability and measure theory:

Measure theory:

Definition:

A set function P defined on a `sigma` -field is called a ‘countablyadditive probability measure is if in addition to satisfying equations , it satisfies the following countable additive property:

for any sequence of pairwise disjoint sets An with A = `uu`nAn
P(A) =`sum` n P(An).

Example 1:

Show that finite additive probability measure P(·) defined
on a σ-field B, is countably additive, that is, if and only
if it satisfies any the following two equivalent conditions.

solution:

If An is any non increasing sequence of sets in B and A = limn An = `nn`nAn
then
P(A) = limn P(An).
If An is any non decreasing sequence of sets in B and A = limn An = `uu` nAn
then
P(A) = limnP(An).

Comprehend more on about Irrational Numbers and its Circumstances. Between, if you have problem on these topics the Poisson distribution Please share your views here by commenting.