MTH 3993, Selected Topics in Biostochastics
Instructors: Eugene H. Dshalalow. Mathematical Sciences, and Richard Sinden, and Robert van
Woesik, Biological Sciences.
Course Outline
This course presents statistical and stochastic tools that are pertinent to biological and ecological
applications. The course will consist of four modules that may vary from year to year depending
on the biological examples considered. The mathematical principles covered will include
probability theory; basic statistics, including maximum likelihood estimates, hypotheses testing,
and simple linear regression. More advanced topics will deal with conditional expectation and its
utility: Bayesian analysis, and Poisson and Markov processes (both in discrete and continuous
time). Biological examples include mutation rates of cancer cells, processes influencing
population growth, instability of DNA repeats associated with neurodegenerative diseases,
stochastic processes in gene regulation and the early immune response.
Course Prerequisites
While the student who enrolls in this course will feel most comfortable with a background in
probability and statistics (like MTH 2401), this is not mandatory and the course self-contained.
However, calculus I-II must precede this course as well as Primer for Biomath.
Course Content Will Include Some of the Following
MODULE I: ELEMENTARY PROBABILITY
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Sample Space. Basic set theory.
Probability and Laplace probability spaces, counting in Laplace probability models.
Conditional probability. Independence. Bayes formula
Discrete random variables [Binomial, Pascal, geometric, Poisson, hypergeometric].
Expectation, variance, and moments.
Continuous random variables. Gaussian, uniform, exponential, and gamma random
variables.
Moments and moment generating functions
Joint Probability Distributions, Marginal, conditional, independence. Covariance.
Sampling. sample mean and sample variance, distribution of sample mean for normal
random variables. Central Limit Theorem. Gamma (revisited). Chi-square distribution.
MODULE II: STATISTICS
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2.
Estimation. Unbiased Estimator. Maximum likelihood estimator, confidence interval.
Determination of mutation rates.
Hypotheses Testing. 1st and 2nd type errors. The power function. Testing simple hypotheses
3.
about the mean, with a known variance. Testing with variance unknown. Goodness-of-fit
tests (Kolmogorov-Smirnov test).
Simple Linear Regression. Inferences concerning regression coefficients. Prediction.
MODULE III: ADVANCED TOPICS IN STATISTICS
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Conditional expectation and conditional variance.
Bayesian analysis. Prior and posterior distributions. Conjugate prior distributions. Bayes
estimators.
MODULE IV: STOCHASTIC PROCESSES WITH APPLICATIONS TO BIOLOGY
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The ordinary Poisson process. Random sums of independent random variables. Marked
Poisson process. Renewal process.
Markov chains. A basic random walk.
Continuous time parameter Markov processes. The Poisson process as a Markov process.
Birth and Death processes, the Yule process.
Applications to Biology. Population growth and population growth subject to random
mutations. Epidemic processes. Processes in gene regulation.
Diffusion processes in genetics.
References
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[2]
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Probability and Statistics, 3rd ed., by M. DeGroot and M. Schervish, Addison Wesley,
2002.
Biostatistical Analysis, 4th ed., by J. Zar, Prentice Hall, 1999.
Fundamentals of Biostatistics 5th ed., by B. Rosner, Duxbury Press, 2000
Probability Models, 8th Ed., by S. Ross, Academic Press, 2006.
An Introduction to Stochastic Processes, by E. Kao, Duxbury Press, 1997
Elements of the Theory of Markov Processes and Their Applications, by Bharucha-Reid,
A.T., McGraw-Hill, New York, 1960.
Stochastic Models with Applications to Genetics, by Tan-Wai-Yuan, World Scientific
Publishers, 2002
Mathematical and Stat Methods for Genetic Analysis, 2nd Ed., by Lange, Springer 2002
GRADING POLICY
Besides the Midterm and Final Exams, all students will take quizzes or turn in their home works
periodically. The final grade will be based on 20% of home work/quizzes, 35% of the midterm
test, and 45% of the final exam.