Probability 2 -- Uncertainty Principle

Theorems in this paper mainly comes from lecture notes from EE503 USC.

Probability 1 -- Sigma Algebra

N Objects Probability Problems

This paper summaries some \(n\) objects probability problems.

Expectation of Stopping Time of Martingale Problems

This paper aims at solving "the expected time until a given pattern occurs" problems.

Stochastic Process 4 -- Martingale

Definition and Theorem in this chapter mainly comes from Stochastic Processes, Second Edition, Sheldon M. Ross

Random Walk Problems

This paper aims at solving all random walk related problems with probability.

Flip Coin Probelms

This paper aims at solving all coin flipping problems with probability.

Stochastic Process 2 -- Poisson Process

Definition: Stopping Time

A stopping time with respect to a filtration \(\{\mathcal{F}_t, t \in \Sigma \subset [0, \infty)\}\) is a mapping \(\tau: \Omega \rightarrow \Sigma \cup \{\infty\}\) such that \(\{\tau \le t\} \in \mathcal{F}_t\) for every \(t \in \Sigma.\)

Stochastic Process 1 -- Preliminaries

Propositions:

1. If \(g: \mathbb{R} \rightarrow \mathbb{R}\) is a measurable function and \(X\) is a random variable, then \(Y = g(X)\) is also a random variable.
2. If \(g\) is a strictly monotone function, let \(g(\mathbb{R})\) be the image of \(\mathbb{R}\) under function \(g.\) Suppose that \(X\) has a continuous density \(f_X.\) Then the random variable \(Y = g(X)\) has density\[ f_Y(y) = \frac{f_X(g^{-1}(y))}{|g'(g^{-1}(y))|}*\mathbb{1}_{g(\mathbb{R})}(y). \]. Here \(\mathbb{1}_{g(\mathbb{R})}\) is indicator function. The formula can be generated to piecewise strictly monotone functions.