STT315: Mathematical Statistics with Applications Department of Mathematics and Statistics, UNC Wilmington Dr. Cuixian Chen 1

What is Statistics?  collection or gathering of data  displaying, analyzing, and summarizing data  inferring information from data

What are the differences b/w STT215 and STT315  STT 315 is more advanced in terms of abstract thinking  STT 315 provides many basic and useful knowledge for advanced statistics classes, as well as classes in other subjects that use statistics.  Actuaries? Data Science? Big Data? Biostatistics?  Stt315 needs solid training in mathematics such as CALCULUS 1,and 2! Skills in derivatives and integrals including multiple integrals are REQUIRED! 3

STT315 Challenges  It is a very difficult class. Some of you may even find it harder than many other graduate statistics courses.  Steep learning curve.  Twisted with probability and Calculus-I/II.  It has its realm of notations and ideas, which may be different from other Mathematical courses.  Some students found out it has nowhere to set out for some probability problems.  No pain, no gain…

STT315 Learning Outcome  We will cover the most fundamental backgrounds for Probability and Statistics.  You will have solid foundation on Statistics and Probability.  You will be well prepared for any advanced graduate level statistics/probability courses.  You are ready to face any challenges on Actuary Exams  Oh… graduate school, I am coming!

DW Simpson 2011 Salary Survey  Source: dwsimpson.com/salary

DW Simpson 2011 Salary Survey  Actuary Exam  "Be An Actuary" Website (click here)Be An Actuary" Websiteclick here  Salary survey for actuary (from DW Simpson) Salary survey for actuary  Actuary Exam P and STT 315 Actuary Exam P and STT 315  Classes offered at UNCW to help you to prepare Actuary Exams.Past Exam Questions and Solutions Classes offered at UNCW to help you to prepare Actuary Exams.Past Exam Questions and Solutions  Exam P/1 Sample Questions and Solutions Sample QuestionsSolutions  MAT381, MAT468 for preparing first two actuary exams.

Courses requirements for Data Science  Biostatistics Analysis/Survival Analysis (STT520/420);  Categorical Data Analysis (STT525/425);  Linear/Multivariate Models and Regression Analysis (STT540/440);  Design of Experiments and Analysis of Variance (STT511/411);  Statistical Inference (STT566/466);  Nonparametric Statistics (STT530/430);  Additional: Probability, Longitudinal Analysis,  Generalized Linear Model;  Introduction to Statistical learning and data mining. STT

Typical Job duty of Biostatistician Duty of Biostatistician:  Write/review Statistical Analysis Plan (SAP).  Write/review statistical section of protocol including sample size calculation, develop randomization schedule, mock up table shells/listing shells, provide specifications to programmers.  Quality control the table from SAS programmer, and review listings.  Validate deliverables. STT

Typical Job duty of SAS programmer Duty for SAS programmer:  Data validation: Data cleaning, Error checking.  Build analysis database, based on the specifications from Biostatistician.  Build table shells and listing shells for the Biostatistician to review and quality control.  Require SAS Macro. STT

Contract Research Organization (CRO) CRO = Contract Research Organization Most projects are the clinical trials from large pharmaceutical company. Local CRO: PPD; Quintiles; INC Research, and so on… STT

Quick review of chapter 1 Sample mean : Sample variance : Sample SD (standard deviation) : Suppose that th observations in a sample are x 1,x 2,…,x n. The sample mean, denoted by is: The sample variance, denoted by s 2, is given by The sample standard deviation, denoted by s, is the positive square root of s 2, that is, The population mean is denoted by µ. The population sd is denoted by σ. HW: EX 1.10,1.11, 1.13 (refer to the observation in 1.12) 12

13 Review STT215 Density Curves and Normal Distribution Definition, pg 56 Introduction to the Practice of Statistics, Sixth Edition © 2009 W.H. Freeman and Company

Normal distributions e = … The base of the natural logarithm π = pi = … Normal – or Gaussian – distributions are a family of symmetrical, bell shaped density curves defined by a mean  (mu) and a standard deviation  (sigma) : N(  ). xx

mean µ = 64.5 standard deviation  = 2.5 N(µ,  ) = N(64.5, 2.5) All Normal curves N  ) share Rule Reminder: µ (mu) is the mean of the idealized curve, while x¯ is the mean of a sample. s (sigma) is the standard deviation of the idealized curve, while s is the s.d. of a sample. About 68% of all observations are within 1 standard deviation (  of the mean (  ). About 95% of all observations are within 2  of the mean . Almost all (99.7%) observations are within 3  of the mean. Inflection point

Standard Normal distributions Empirical Rule: For a distribution of measurements that is approximately normal, it follows that the interval with end points About 68% of all observations are within 1 standard deviation (  of the mean (  ). About 95% of all observations are within 2  of the mean . Almost all (99.7%) observations are within 3  of the mean . N(0,1) => N(64.5, 2.5) Standardized height (no units)