L5 Expectations, Moments, & MGF

Verily, in the creation of the heavens and earth, and the alternation of the night and the day, and the ships which sail through the sea with benefits for people, and what Allah has sent down from the heavens of rain, giving life thereby to the earth after its lifelessness and dispersing therein every creature, and in His directing of the winds and the clouds controlled between the heaven and the earth are signs for a people who use reason.

Surat Al-Baqarah 2:164

Reason is the source and fountainhead of knowledge, as well as its foundation. Knowledge sprouts from it as the fruit does from a tree, as light comes from the sun, and as vision comes from the eye. How then could that which is the means of happiness in this life and the Hereafter not be considered the most honored? Or how could it be doubted?

Source: Al-Ghazali: Iḥyāʼ Ulūm al-Dīn 1/83

LECTURE 5: Expectations, Moments, and Applications

Suppose X is a Random Variable, and g(X) is a real valued function of X. In the pre-experimental period, what do we expect to see as the value of g(X)? Pre-Experimentally the actual value of X has not been observed, and all possible outcomes are present as potential outcomes. The EXPECTATION of g(X) evaluates g(X) at all of the potential outcomes, weighting each outcome according to its probability of occurrence. Thus it calculates the average value of g(X) in a perfect and imaginary world, where each outcome occurs precisely according to its potential probability for occurrence. This PRE-EXPERIMENTAL quantity has a POST-EXPERIMENTAL analog. Given a sequence of OBSERVED IID outcomes of X, X1, X2, ..., Xn, we can calculate the actual observed average value of g(X) as the average of the observations g(X1), g(X2), ... , g(Xn). This is the AVERAGE or MEAN of g(X) on a sequence of IID observations of X. The POST EXPERIMENTAL averages converge to the PRE-EXPERIMENTAL ideal in large samples. This theory is central to probability and statistics as it establishes the correspondence between the imaginary world of probabilities and the real world of observations.

C1 Expectations

C2 Moments

C3 Moment Generating Functions

C4 Sums of IID Random Variables

C5 Law of Large Numbers

Abū ʿAlī al-Ḥasan ibn al-Ḥasan ibn al-Haytham (Arabic: أبو علي، الحسن بن الحسن بن الهيثم‎‎;Persian: بوعلی محمد بن حسن بن هیثم‎‎ c. 965 – c. 1040 ce), also known by the Latinization Alhazen or Alhacen,^[10] was an Arab^[11][12] Muslim^[13][14][15]scientist, mathematician, astronomer, and philosopher.^[16] Ibn al-Haytham made significant contributions to the principles of optics, astronomy, mathematics,visual perception, and the scientific method.^[17] He was the first to explain that vision occurs when light bounces on an object and then is directed to one's eyes.^[18] He spent most of his life close to the court of the Fatimid Caliphate inCairo and earned his living authoring various treatises and tutoring members of the nobilities.^[19]

Ibn al-Haytham is widely considered to be one of the first theoretical physicists, and an early proponent of the concept that a hypothesis must be proved by experiments based on confirmable procedures or mathematical evidence—hence understanding the scientific method 200 years before Renaissance scientists.^{[20][21][22][23][24][25]}

In medieval Europe, Ibn al-Haytham was honored as Ptolemaeus Secundus(the "Second Ptolemy")^[26] or simply called "The Physicist".^[27] He is also sometimes called al-Baṣrī after his birthplace Basra in Iraq,^[28] or al-Miṣrī ("of Egypt").^[29]