Applications

-
Applications

Commenti

Posta un commento

Post popolari in questo blog

Welford Algorithm

-
Immagine
The variance is the expectation of the squared deviation of a random variable form its mean.  Informally, it measures how far a set of (random) numbers are spread out from their average value. The variance is the square of the standard deviation, the second central moment of a distribution and the covariance of the random variable with itself, and it is often represented by Var(X), σ 2 , s 2 . One of the most widely known formula for computing the variance is: where x-bar is the mean of the sample. The definition given above can be converted into an algorithm that computed the variance and the standard deviation in two passes: 1. Compute the mean                            (O(n)) 2. Compute the square differences       (O(n)) Output the variance Even though this algorithm seems working properly, it may become too expensive on some input instances. Just consider a sampling procedu...
Continua a leggere

Insight 3 - Bayes' Theorem

-
Before giving the Bayes theorem statement is important to define the conditional probability. Once done, the Bayes theorem is easy to derive. Conditional Probability The conditional probability measures the likelihood of an event given that another event has occurred. For instance, assuming that  A  is the event of interest and that  B consists of a different event which has already occurred. The probability of the event  A  to occur, given the occurrence of B , (written as)  P ( A  | B ), can be computed as detailed by the following equality. P(A | B) = P(A ∩ B) / P(B)        P( B ) > 0                               (1) It is defined as the quotient of the probability of the joint events A  and B  and the probability of B . Now consider P(B | A), which is equals to P(B ∩ A) / P(A). Since P(B ∩ A) =  P(A ∩ B), then the fo...
Continua a leggere

Insight 2 - Kolmogorov Axioms and Relative Frequency

-
Immagine
Probability theory is a branch of mathematics concerned with the analysis of phenomena, characterised by randomness or uncertainty, in order to model/predict the behaviour of defined systems. Although there are several different probability interpretations, such as the propensity interpretation or the subjective one, the most used and established probability theory is due to Andrey N. Kolmogorov, a Russian mathematician which combined previous studies on the field and presented his axiom system for probability theory in 1933. This blog post is intended to introduce the reader to the main axioms and rules regarding the Kolmogorov's theory. Let S  denote a sample space with a probability measure P  defined over it, such that the probability of any event E ⊂ S  is given by P(E) . Then, the probability measure obeys the following axioms: AXIOM 1 - The probability of an event is a non-negative real number: P(E)  ∈   ℝ,   P(E) ≥ 0   ...
Continua a leggere