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Introduction_to_Probability-The_Science_of_Uncertainty_NOTES.txt
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Introduction_to_Probability-The_Science_of_Uncertainty_NOTES.txt
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a probabilistic model is a quantitative description of a situation / phonaminon / experement where whose outcome is uncirtien.
a probabilistic model involves these steps:
1- Describe set pf possible outcomes.
This set must be:
- Mutually exclusive.
at the end there is exactly 1 out come.
- Collectively exhaustive.
is set me sary possible outcomes hon.
- At the "right" granularity.
eg: agar mujh ye dekhna ho k coin fear h ya nahi, to mera output set ye hona chaye {H,T}, or ye nahi hona chahye {H_and_rain, H_not_rain, T_rain, T_not_rain}
2- Describe belifes about likelihood of outcomes.
3- identify an event of interest
4- calculate
whenever we have an experement with several stages (like : probability of getting 3 head in rowlling 3 dies) it is very usefull way of describing it is by providing a squential description in terms of tree.
sample space:
can be infinite, and can be continues.
1- discrete and finite:
eg: getting two head in rolling 2 coins.
2- continues and finite:
eg: aap mahir teer andaz hen, or har bar aap ka teer nishany (1 x 1 fit) k andar lagta h, ab aap ny kafi dafa teer chalay, or har dafa jahan aap ka teer laga h us ka x,y cordinates record kar lye. to aap ka sample space yahan 0 <= x, y <= 1 ho ga, jo continues or finite h.
The ONLY probability axioms:
1- non negativity
is axiom sy 1 or axiom bhi sabit hota h k p(A) <= 1
proof: p(A) = 1 - p(not_A), and p(not_A) >= 0. so p(A) must be <= 1
2- notmalization
probabiliry(set(all_possible_outcomes)) = p(A union not_A) = 1
is axiom sy 1 or axiom bhi sabit hota h k P(empty_set) = 0
proof #1 :
since A union not_A = entire_set ==>
A intersection not_A = empty_set
prrof #2:
1 = p(all_possible_outcomes) + p(complement_of_all_possible_outcomes) ==>
1 = 1 + p(complement_of_all_possible_outcomes) ==>
p(complement_of_all_possible_outcomes) = 0
for disjoint events:
3- (finite) additivity
only for countable sequence of events, hence it is for descrete but not for continues. so unit square and read line etc is not countable.
"Area" is lagitimate probability law on the unit square, as long as we do not try to assing probabilites/areas to "very strange" sets.
if A and B are disjoint then p(A union B) = p(A) + p(B)
genralization of additivity axiom:
if A,B and C are disjoint, then p(A union B union C) = p(A) + p(B) + p(C)
sample = {A,B,C,D}
set_of_interest = {A,B,C}
sum([p(k) for k in set_of_interest])
proof:
p(A union B union C) =
p( p(A union B) union C) =
p(A union B) + p(C)
some consequences of the probability axioms:
if A is the part of B, then p(A) <= p(B).
proof:
B = A union (B intersection not_A) ==>
since B and not_a are disjoint we apply additivity axiom here
p(B) = p(A) + p(B intersection not_A) ==>
since <p(B intersection not_A)> >= 0;
p(B intersection not_A) >= p(A)
if A and B are not necessery disjoint:
p(A union B) =
p(A) + p(B) - p(A intersection B)
anoter way:
p(A instersection not_B) + p(B intersection not_A) + p(A instersection B)
if A,B and C not necessery disjoint:
p(A union B union C):
p(A) + p(not_A intersection B) + p(not_A intersection not_B intersection C)
p(A union B) <= p(A) + p(B)
probability for continues:
ksi bhi infinite precistion ki probability (almost) 0 hoti h, isi wajah sy Event (a subset of a sample space) ki probaility lety hen, eg: jab hame let sy pr(age==66) chahye hota h to darasal ham is cheez ki probability dekhty hen <pr(age >65) & pr(age <= 66)>
intersection <n>:
in both
order does not metter, so A intersection B intersection C = B intersection A intersection C
union <U>:
in any of both A and B
order does not metter, so A union B union C = B union A union C
we can define unions and intersections of more than 2 sets, even of infinitly meny sets.
SET:
A collection of distinct elements.
it is finite (eg:{a,b,c}) or infinite (eg: real numbers).
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Conditioning_and_independence:
probaility is a way of descibing our belives about the likelihod the the given event will occur. our belivies will in general depend on the information that we have. taking into account new information lives as to consider so colled conditnal probabilites, these are reviesed probabilites that take into account the new information.