This document is a compilation of my notes for MTH 514: Probability and Stochastic Processes at TMU. All information comes from my professor’s lectures, as well as the course textbook Introduction to Probability, Statistics and Stochastic Processes, H. Pishro-Nik.

Adam Szava - 2tor.ca

F2022

Chapter 1: Basic Concepts

For notes on this topic, refer to the first chapter of my MTH 410: Statistics notes available at www.2tor.ca/notes

Chapter 2: Combinatorics

For notes on this topic, refer to the first chapter of my MTH 410: Statistics notes available at www.2tor.ca/notes

Chapter 3: Discrete Random Variables

For notes on this topic, refer to the second chapter of my MTH 410: Statistics notes available at www.2tor.ca/notes

Chapter 4: Continuous and Mixed Random Variables

For notes on this topic, refer to the third chapter of my MTH 410: Statistics notes available at www.2tor.ca/notes

Any notes in this section are just additional interesting things from the textbook/lecture.

4.1.1: Probability Density Functions (PDFs)

The PDF of a continuous R.V. can also be interpreted in the following way:

If $f_X(x_1)>f_X(x_2)$, we can say that $P(x_1<X\leq x_1 + \delta)>P(x_2<X\leq x_2 + \delta)$ for some small $\delta.$ This means the value of $X$ is more likely to be around points like $x_1$ which have a relatively higher $f_X(X =x_1)$ values.

Chapter 5: Joint Distributions

In real life, we often are interested in studying several random variables at the same time, each of which measure some quantity of the random experiment. In this chapter we look at how we can study two discrete R.Vs and two continuous R.Vs.

Extending this to $n$ R.Vs of the same type is then easy.

5.1.1 Joint Probability Mass Function (PMF) (Discrete)