Probability MITx 6.041x Notes
Last Updated: September 09, 2021 by Pepe Sandoval
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Quantitative description: it is a description in terms of numbers
A probabilistic model is a quantitative description of a situation, a phenomenon, or an experiment whose outcome is uncertain.
set
) of the possible outcomes ($\Omega$), the elements of the set must be:$$A \subset B \Rightarrow P(A) \le P(B)$$
$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
Probability calculations steps:
There are two kinds of infinte sets: 1) discrete (countable) which are set whose elements can be arranged in a sequence like the integers and 2) uncountable sets whose elements cannot be arranged in a sequence
Probabilities can be interpreted as frequencies: the frequency with which certain event will occur in a infinite number of repetitions of the experiment
Probabilities are oftenly used informally to decribe beliefs or likelihood of events
Probability gives a set of rules to think systematically about uncertain situations which can be used as tool to make predictions and decisions
Real World generates data, statistics uses this data to come up with probabilistic models, we use probaility theory and this model to make predictions and desicions about the real world
A set is a collection of distinct elements, that can be infinite or inifinite
Sets operations
$$(S \cap T)^c = S^c \cup T^c$$
$$(S \cup T)^c = S^c \cap T^c$$
A sequence is a collection of elements that are coming out of some set and that collection of elements is indexed by the natural numbers (positive integers)
Formally it is a function that to any natural number assigns an element of a set ($S$)
We usually want to know if a sequence converge, this means we whant to know if as the index tends to infinity the sequence tends to an specifici number.
Formally what it means that a sequence converge is that if we plot the sequence as a function of $i$ as $i$ tends to infinity the values of $a_i$ get and stay inside a small band (defined by \epsilon)
if a sequence converge we can assume certain properties like the multiplication, sum or application of a function can be applied to the value the sequence(s) converge
$$ \sum_{i=1}^{\inf} a_i =\lim_{n\rightarrow\infty} \sum_{i=1}^n a_i $$
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