Introduction

Several terms need to be clar­i­fied math­e­mat­i­cally to ap­pre­ci­ate the large sets of data which are typ­i­cal of ex­per­i­ments. Their rel­e­vance lies in the fact that ex­trap­o­la­tion of math­e­mat­i­cal terms is eas­ier than quan­ti­fy­ing in words, more com­plex sys­tems. Where ap­plic­a­ble, gen­eral in­ter­pre­ta­tions are in­di­cated, how­ever, most of the dis­cus­sion of these terms is left to the sec­ond chap­ter; where spe­cific chem­i­cal en­gi­neer­ing ap­pli­ca­tions are enu­mer­ated.

We seek to first place on an math­e­mat­i­cal foot­ing, the eas­ily grasped con­cepts of var­i­ous ex­per­i­men­tal terms (e.g., ran­dom vari­ables). Subsequently it is shown via sim­ple math­e­mat­i­cal con­structs, how prior con­cepts may be gen­er­al­ized (moments). Finally the util­ity of these con­structs is ex­plored briefly in con­text of pre­dic­tive abil­ity and the sys­tem­atic treat­ment of non-in­tu­itive re­sults.

Elementary set the­ory is a pre-as­sumed, and cer­tain as­pects re­quire a level of math­e­mat­i­cal rigor best ob­tained by a fa­mil­iar­ity with the rigid­ity of tech­ni­cal de­f­i­n­i­tions.

The ap­par­ent ab­stract­ness of this sec­tion is off­set by the util­ity of hav­ing math­e­mat­i­cal con­cepts ac­ces­si­ble with­out be­ing tied to phys­i­cal ex­am­ples. (i.e., al­low­ing con­cepts to re­tain a log­i­cal pu­rity.)

Additionally, the study of mo­ments forms the cor­ner­stone of a gen­er­al­ized study of sta­tis­tics in it’s en­tirety. Recognizing this, fo­cus has been given to mainly the ini­tial and cen­tral mo­ments of sin­gle ran­dom vari­ables in this trea­tise.

This col­lec­tion is largely in­spired from Bronshtein et al. (2015), Polyanin and Manzhirov (2006) and Polyanin and Chernoutsan (2010).

Articles

This se­ries con­sists of:

• Mathematical pre­lim­i­nar­ies
• Moments and their in­ter­pre­ta­tions

References