This book provides a primer in inequalities in applied probability theory and statistics. It is intended to be useful to both graduate students and established researchers working in Probability Theory and Statistics, Analytic Integral Inequalities and their applications in demography, economics, physics, biology and other scientific areas. The book is self-contained in the sense that the reader needs only to be familiar with basic real analysis, integration theory and probability theory. All inequalities used in the text are explicitly stated and appropriately referenced.