MATH385 — Probability and Statistical Inference I
Mathematical theory of probability and statistical inference: proofs, discrete/continuous random variables, sampling distributions, central limit theorem.
Learning Objectives & Matches
Use the multiplication rule to derive the number of permutations of r objects chosen from n and the number of combinations of r objects chosen from n
Apply sampling techniques to determine groups to be surveyed or use complete enumeration methods.
Apply sampling techniques, or use complete enumeration bases to determine and define groups to be surveyed.
Plan data collection methods for specific projects, and determine the types and sizes of sample groups to be used.
Calculate sample size requirements for clinical studies.
Determine whether statistical methods are appropriate, based on user needs or research questions of interest.
Develop and test experimental designs, sampling techniques, and analytical methods.
Prove probability theorems using rigorous mathematical methods
Construct probability tables for events such as fires, natural disasters, and unemployment, based on analysis of statistical data and other pertinent information.
Compute the number of permutations of r objects chosen from n and the number of combinations of r objects chosen from n
Apply probability distributions to model random phenomena
Examine theories, such as those of probability and inference, to discover mathematical bases for new or improved methods of obtaining and evaluating numerical data.
Adapt statistical methods to solve specific problems in many fields, such as economics, biology, and engineering.
Construct probability tables for events such as fires, natural disasters, and unemployment, based on analysis of statistical data and other pertinent information.
Assemble sets of assumptions, and explore the consequences of each set.
When presented with a word problem, decide whether permutations or combinations are necessary and use the appropriate tool(s) for counting the number of sample points in an event and in the sample space
Apply sampling techniques, or use complete enumeration bases to determine and define groups to be surveyed.
Apply sampling techniques to determine groups to be surveyed or use complete enumeration methods.
Determine whether statistical methods are appropriate, based on user needs or research questions of interest.
Calculate sample size requirements for clinical studies.
Collect data through surveys or experimentation.
Develop estimators and analyze their properties
Recognize the need for Bayes' rule in a word problem, will be able to use it and will be able to prove it
Perform hypothesis testing using classical and modern methods
Document test plans, testing procedures, or test results.
Evaluate the statistical methods and procedures used to obtain data to ensure validity, applicability, efficiency, and accuracy.
Perform Web site tests according to planned schedules, or after any Web site or product revision.
Perform Web site tests according to planned schedules, or after any Web site or product revision.
Monitor clinical trials or experiments to ensure adherence to established procedures or to verify the quality of data collected.
Calculate sample size requirements for clinical studies.
Collect data through surveys or experimentation.
Determine whether statistical methods are appropriate, based on user needs or research questions of interest.
Draw conclusions or make predictions, based on data summaries or statistical analyses.
Test programs or databases, correct errors, and make necessary modifications.
Work word problems which require the ability to recognize the probabilities and conditional probabilities which are being given and being requested
Formulate mathematical or simulation models of problems, relating constants and variables, restrictions, alternatives, conflicting objectives, and their numerical parameters.
Conduct logical analyses of business, scientific, engineering, and other technical problems, formulating mathematical models of problems for solution by computers.
Assign or schedule tasks to meet work priorities and goals.
Assign work to biostatistical assistants or programmers.
Adapt statistical methods to solve specific problems in many fields, such as economics, biology, and engineering.
Develop computational methods for solving problems that occur in areas of science and engineering or that come from applications in business or industry.
Provide staff and users with assistance solving computer-related problems, such as malfunctions and program problems.
Collaborate with senior managers and decision makers to identify and solve a variety of problems and to clarify management objectives.
Confer with project personnel to identify and resolve problems.
Analyze information obtained from management to conceptualize and define operational problems.
Construct confidence intervals for population parameters
Calculate sample size requirements for clinical studies.
Calculate sample size requirements for clinical studies.
Report results of statistical analyses, including information in the form of graphs, charts, and tables.
Prepare tables and graphs to present clinical data or results.
Determine whether statistical methods are appropriate, based on user needs or research questions of interest.
Evaluate the statistical methods and procedures used to obtain data to ensure validity, applicability, efficiency, and accuracy.
Determine whether or not a function is a probability function
Apply probability theory to real-world decision making
Find the mean variance of discrete and continuous random variables, including instances when such determinations require the use of geometric series, Taylor series, improper integrals and/or integration by parts
Use moment generating functions and characteristic functions
State and apply the Central Limit Theorem
Given that X has a normal distribution with mean m and variance s^2, derive the distribution of (X - m)/s. Use this result along with normal probability tables to calculate probabilities associated with X
Set-up the appropriate null and alternative hypotheses from word problems where one- or two-sided one-sample tests of population means are indicated. Determine which hypothesis is which and whether or not the alternative is one- or two-sided
Specify inputs accessed by the system and plan the distribution and use of the results.
Apply mathematical theories and techniques to the solution of practical problems in business, engineering, the sciences, or other fields.
Confer with clients regarding the nature of the information processing or computation needs a computer program is to address.
Collaborate with others in the organization to ensure successful implementation of chosen problem solutions.
Examine theories, such as those of probability and inference, to discover mathematical bases for new or improved methods of obtaining and evaluating numerical data.
Analyze problems to develop solutions involving computer hardware and software.
Specify manipulative or computational methods to be applied to models.
Calculate confidence intervals for the mean of a normal population when the population variance is known and when it is not, and understand when to use the t-tables in the calculation of confidence intervals and when to use normal tables