### Probability & Statistics for BME

#### BME 32200 / 3 Cr.

BME 32200 is a combined lecture and laboratory course that provides an introductory treatment of probability theory, including distribution functions, moments, and random variables. Practical applications include; estimation of means and variances, hypothesis testing, sampling theory and linear regression. The application of normal and exponential distributions in the statistical analysis of biological variables is covered extensively. Introduction to random processes, correlation functions and spectral density functions are also covered.

**Available Online:** No

**Credit by Exam:** No

**Laptop required:** No

**Prerequisites/Co-requisites:**

P: BME 33100 and BME 33400. C: None.

##### Textbooks

Intuitive Probability and Random Processes using MATLAB by Steven Kay (2006), Springer. ISBN: 0-387-24157-4. Both electronic and printed handouts will also be distributed throughout the semester.

##### Goals

This course provides the foundational skills for advanced statistical analysis of biological signals. The basic analytical concepts of probability theory, statistical design of experiments and data analysis and representation of biological variables as random processes are demonstrated and practiced through computer based analysis of biologically relevant data sets provided throughout the course. All computational homework assignments are carried out using MATLAB. All laboratory exercises are carried out using LabVIEW.

##### Outcomes

Upon completion of the course, students should be able to:

- Solve simple probability problems with electrical and computer engineering applications using the basic axioms of probability. [a, e]
- Describe the fundamental properties of probability density functions with applications to single and multivariate random variables. [a, b2, e]
- Describe the functional characteristics of probability density functions frequently encountered in life science research such as the Binomial, Uniform, Gaussian and Poisson. [a, b2]
- Determine the first through fourth moments of any probability density function using the moment generating function. [a, e]
- Calculate confidence intervals and levels of statistical significance using fundamental measures of expectation and variance for a given numerical data set. [b2]
- Discern between random variables and random processes for given mathematical functions and numerical data sets. [a, b2]
- Determine the power spectral density of a random process for given mathematical functions and biological data sets. [a, b2]
- Determine whether a random process is ergodic or nonergodic and demonstrate an ability to quantify the level of correlation between sets of random processes for given mathematical functions and biological data sets. [a, b2]
- Model complex families of signals by means of random processes. [a]
- Determine the random process model for the output of a linear system when the system and input random process models are known. [a, c, e]

##### Topics

BME 32200 is comprised of three interrelated subject areas, all involving the use of mathematical and computational tools to distill biological data into meaningful statistical representations. The first subject area broadly introduces the topic of probability theory (e.g. relative-frequency, set theory, and axioms of probability, conditional probability, independence, and Bernoulli trials) as related to sampled data. This leads to the introduction of random variables and distribution functions (e.g. probability density functions, mean values and moments, Gaussian random variables, density functions conditional density functions, joint distributions, covariance, sums of random variables) along with sampling and estimation theory (e.g. point and interval estimation, sampling distributions, estimation of means and variances, hypothesis testing, regression analysis and goodness-of-fit tests). The second subject area focuses upon random process definitions and measurement of random processes from biological signal sources (e.g. correlation, cross-correlation and applications to analysis of random processes from multiple biological sources). The third subject area utilizes recent articles from the scientific literature demonstrating the application of these and other mathematical processing techniques ( e.g. spectral density, properties of spectral density, and mean-square values from spectral density) in the study of biological signals and physiological systems. Refer to the lecture schedule for specific topics and dates.

##### Additional Reference Materials

- Probabilistic Methods of signal and System analysis by Cooper & McGillem (1999), Oxford University Press. ISBN: 0-190512354-9
- Probability and Random Processes by Childers (1997), McGraw-Hill. ISBN: 0-256-13361-1
- Probability and Statistics for the Engineering, Computing and Physical Sciences by Dougherty (1990), Prentice Hall. ISBN: 013711995X
- Principles of Neuroscience by Kandel & Schwartz (2002), Elsevier. ISBN: 0838577016
- Physiology by Berne (Editor), Levy, Koeppen & Stanton (2002), Mosby. ISBN: 0815109520
- Biomedical Engineering Handbook 2 nd edition by Bronzino (Editor), CRC Press, ISBN 0849383463