## Math 310 & Stat 410 at Lehigh University

Math 310/Stat 410 (Probability and Its Applications): Random variables, moment-generating and characteristic functions; limit theorems; stochastic processes, Kolmogorov euqations; Markov chains, random walks. Prerequisite: Math 309 or consent of the department chairperson. (Poisson processes and applications on queuing theory and reliability theory are included in the course. Brownian motion and stationary processes and applications to pricing stock options are typically included in the course.)

Here is a typical course description.

This course can be taken as a 400-level (3 graduate credits).

The text is `Introduction to probability Models', 6th edition, by S. Ross.

Topics to be covered:

• Review (sample space, probability measure, conditional probability, independence, randon variables, some known discrete and continuous distributions, joint distributions, expected values, variance, covariance).
• Moments and Moment generating function.
• Can moments always uniquely determine a probability distributions?
• The moment-generating technique
• Limit theorems (Convergence in probability, LLN, Central Limit Theorem, and convergence in distribution).
• Markov's Inequality; Chebyshev's Inequality; Boole's Inequality;
• Conditional probability and conditional expectation
• Computing probabilities by conditioning
• Computing expectations by conditioning
• Recursive equations and the method of conditioning
• Markov chains
• Transition probabilities
• Hitting times
• Chapman-Kolmogorov equations
• Classification of states
• Induced martingales
• Birth and death chains
• Limiting and stationary probabilities of markov chains
• Exponential and Poisson distributions
• Poisson process
• Counting process
• Interarrival and waiting time distributions
• Nonhomogeneous and compound processes
• Birth and death process
• The Kolmogorov differential equations
• Time reversibility
• Uniformization
• Renewal process
• Some applications in queueing theory
• Random walks and Brownian motion
• White noise
• Simulation
• Other special topics