Introduction to likelihood and measure theory:
Measure concept:
Step theory is a branch of mathematical evaluation which studies the concept of size of an abstract collection and accumulations of a function. Action concept, to a big extent, is self contained, but initial concepts on set theory, features and real evaluation is crucial. Probability and step theory:
Chance:
Example 1:
Consider two similar coins. What is the possibility that the coins suit (both heads or both tails)?
Solutions:
If we throw two identical coins the set of feasible end results is S = HH, HT, TT. However, these easy occasions are not just as most likely; there is only one method to obtain 2 heads and only one way to obtain 0 heads, yet exactly 1 head can happen in two methods. If the coins were appreciable, after that
the sample area would be
R = HH, HT, TH, TT
and these events are equally most likely with each having the probability 1/4. What is the probability that at least one head shows up?
Option:
If the coins were distinguishable, after that the sample space would be
R = HH, HT, TH, TT
and these events are similarly likely with each having the possibility 1/4. Keep in mind that the compound occasion HT, TH in R represents the simple occasion HT (or specifically 1 head) in S. Instance 1:
Program that limited additive probability procedure P( ·) defined
on a σσ-field B, is countably additive, that is, if and only
if it pleases any the following 2 comparable problems. If An is any non decreasing series of sets in B and A = limn An='uu' nAn
then
P(A) = limnP(An).