Here are my notes for E, Li, and Vanden-Eijnden’s Applied Stochastic Analysis Chapter 5 - Stochastic Processes 5.1 - Axiomatic Construction of Stochastic Process 5.2 - Filtration and Stopping Time 5.3 - Markov Processes 5.4 - Gaussian Processes Chapter 6 - Wiener Process 6.1 - The Diffusion Limit of Random Walks 6.2 - The Invariance Principle
I was recently doing a probability puzzle that I can’t quite remember the context of, but I came across the answer that the probability would be $$\mathbb{P}(X) = n p^n \; \quad \forall \: n\in\mathbb{N}, p \in [0,1].$$But this is obviously wrong! Plug in $p=.9, n=2$, and you get that $\mathbb{P}(X) = 1.62$. Thaat’s not how probability works! However, for $p=0.5$, $\mathbb{P}(X)$ will remain $\leq 1$ for all $n \in \mathbb{N}$. So, somewhere in the interval $(0.5,0.9)$, we reach a critical value where any $p$ greater than that will result in a probability greater than one, and any value less than it will be a bit more reasonable. ...