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:tada: PROBABILITY AND MONTE CARLO :tada: (What is a Random Experiment? …
:tada: PROBABILITY AND MONTE CARLO :tada:
What is a Random Experiment? :silhouettes:
Same experiment => different result each time.
Set of possible results is the
sample space
/Omega.
Discrete
Sample space: finite, countably infinite outcomes
Continuous
sample space: an interval of real numbers, vectors
A
subset
of sample space of a random experiment is an
event
:red_flag:
UNION of two events: A or B :star:
INTERSECTION of two events: A and B :star:
if A & B = NULL, then
mutually exclusive
COMPLEMENT of an event: Not A :star:
logical laws
Distributive law
p∨(q∧r)≡(p∨q)∧(p∨r)
De Morgans Law
¬(a∨b)=(¬a)∧(¬b), or alternatively ¬(a∧b)=(¬a)∨(¬b).
ie, an event is an element in the POWER SET of the sample space set
What is
pseudorandomness
?
a recursively defined sequence with a well defined functional relationship. begins with the
seed
(U_0) and for any seed U_i = U_k, the sequence is exactly repeatable. the sequence is usually then normalised to get in the interval(0,1)
a good sequence:
computationally easy to get the next element
uniform distribution, expected value is 0.5
Random experiments are often performed in
simulations
using the
monte carlo
method which uses this seed + functional formula combo to generate pseudorandom numbers
What is Probability? :pencil2:
A probability is
a way of assigning
every event a value between zero and one
1) requirement that the event made up of all possible results (in our example, the event {1,2,3,4,5,6} for a regular die) is assigned a value of one
TIB each element in that event has a probability of 1/N, so the sum of the probabilities of N events is 1.
2) requirement that if you look at a collection of mutually exclusive events, the probability that
at least one
of the events will occur is given by the sum of the probabilities of all the individual events.
conditional probability
P(A|B) is the probability of some event A, given the occurrence of some other event B.
P(A|B) = P(A&B) / P(A) for P(A) > 0
probability laws
Multiplication Rule
When we calculate probabilities involving one event AND another event occurring, we multiply their probabilities.
General Statement :!!:
P(A&B)=P(A)⋅P(B∣A)
Law of Total Probability
fundamental rule relating
marginal probabilities
to
conditional probabilities
.
Given n mutually exclusive events A_1, ..., A_n whose probabilities sum to unity, then
P(B)=P(B|A_1)P(A_1)+...+P(B|A_n)P(A_n)
,
where B is an arbitrary event, and P(B|A_i) is the conditional probability of B assuming A_i.
Independent Probability
Conditions:
(1) P(A|B) = P(A)
(2) P(B|A) = P(B)
(3) P(A&B) = P(A) P(B)
this extends to multiple events as well
INDEPENDENT means something totally different from MUTUALLY EXCLUSIVE