Download Applied Probability at the Sophomore Level:

What should students learn, and when?
Matthew A. Carlton
Statistics Department
California Polytechnic State University
San Luis Obispo, CA, USA


Primary references (yes, first!)
2014 ASA Guidelines: Probability/Math Stat
◦
◦
◦
◦

Philosophy
Content
Computing
Timing
Two curricular models
◦ BYU (very brief)
◦ Cal Poly


“Curriculum Guidelines for Undergraduate
Programs in Statistical Science,” ASA, 2014.
Horton, N., “The increasing role of data
science in undergraduate statistics programs:
new guidelines, new opportunities, and new
challenges,” webinar, Feb. 3, 2015.


“[Stat majors] need a foundation in
theoretical statistical principles for sound
analyses.” (p. 9)
“[Stat majors] should have a firm
understanding of why and when statistical
methods work.” (p. 12)


Statistical theory includes: “distributions of random
variables, likelihood theory, point and interval
estimation, hypothesis testing, decision theory,
Bayesian methods, and resampling” (p. 11)
Mathematical foundations include: “probability (e.g.,
univariate and multivariate rvs, discrete and continuous
distributions)”; “emphasis on connections between
concepts … and their applications in statistics” (p. 12)


“Theoretical/mathematical and computational/
simulation approaches are complementary, each
helping to clarify understanding gained from the
other.” (p. 8)
“[Stat majors] should be able to … use simulationbased statistical techniques and to undertake
simulation studies.” (p. 9)


“If included early on in a student’s program,
[probability and math stat] will help provide a solid
foundation for future courses and experiential
opportunities.” (p. 16, emphasis added)
Countervailing force: math prerequisites (p. 16)




Probability/math stat is foundational
Material needs to connect to applications
Include simulation; don’t divorce probability
from technology
Present probability/math stat earlier, so they
can be leveraged later (but, again, math
prereqs can be an obstacle)

BYU model (very brief)
◦ Based on email discussions with BYU faculty

Cal Poly model

STAT 240: Discrete Probability
◦ Sophomore year, first term
◦ Prerequisite: one previous statistics course
◦ Text: Goldberg (1960)

STAT 340: Inference
◦ Junior year, first term
◦ Prerequisite: STAT 240, Calculus II
◦ Text: DeGroot & Schervish (2011)

STAT 305: Intro to Probability & Simulation
◦ Sophomore year, first term
◦ Prerequisites:
 Calculus II
 a computer programming course
◦ Text: Carlton & Devore (2014)

STAT 305: Course objectives
1. Use definitions, rules, and counting methods to
solve probability problems
2. Calculate probabilities, expected values, and
variances related to discrete and continuous rvs
3. Identify and apply probability distributions to
solve probability problems
 Emphasis on applications, not “proof-oriented”

STAT 305: Course objectives
4. Apply properties of expected values and variances
to linear combinations of random variables
 Not proof- or derivation-based
 Focus on applications to statistical estimators,
especially standard deviation of linear combinations
 Sets the stage for junior- and senior-level electives

STAT 305: Course objectives
5. Simulate random phenomena to approximate
probabilities, expected values, and distributions
of random variables
•
•
•
•
•
Integrated throughout course, incl. example code
Emphasize looping (simulation through repetition)
Emphasize measuring uncertainty
Easier/“confirmatory” exercises in homework
Harder/“exploratory” assignments (longer)

STAT 425-6-7: Probability Thy/Math Stat
◦
◦
◦
◦

Junior year, year-long
Prerequisite: Calculus IV, Methods of Proof
Text: DeGroot & Schervish (2011)
Course is also taken by math Master’s students
Elective course: advanced models (Markov
chains, Poisson processes, etc.)



“Non-proof” probability can (and should) be
introduced at the sophomore level.
Students should experience real computer
programming in a probabilistic (i.e., not data
management/analysis) setting.
Math stats can be taught junior year.

[email protected]