ISMT
Bayesian inference and time series analysis
Bayesian inference and time series analysis
Kartei Details
| Karten | 14 |
|---|---|
| Sprache | English |
| Kategorie | Mathematik |
| Stufe | Universität |
| Erstellt / Aktualisiert | 02.10.2022 / 05.02.2024 |
| Lizenzierung | Kein Urheberrechtsschutz (CC0) |
| Weblink |
https://card2brain.ch/box/20221002_ismte136
|
| Einbinden |
<iframe src="https://card2brain.ch/box/20221002_ismte136/embed" width="780" height="150" scrolling="no" frameborder="0"></iframe>
|
[Statistics]
Given
\(E[X]=\sum_i x_i f(x_i)\)
\(E[X^2] = \text{?}\)
\(E[X^2]=Var[X]+E[X]^2\)
We can see that \(E[X^2] \neq E[X]^2 \text{ if } Var[X]\neq 0\)
[Time series analysis]
If AR(1) is causal and given as
\(x_t=\phi x_{t-1}+w_t\text{, where }w_t\sim wn(0, \sigma_w^2)\)
\(Var[x_t]= \text{?}\)
\(\begin{align} &Var[x_t]=var(\alpha + \phi x_{t-1}+w_t) \\&= 0 + var(\phi x_{t-1}) + var(w_t) \\&= \phi^2var(x_{t-1})+\sigma_{w}^2 \\&= \phi^2var(x_{t})+\sigma_{w}^2 &\text{| as series is stationary} \\&= \frac{\sigma{_w}^2}{1-\phi^2}\end{align} \)
[Statistics]
\(cov(aX + bY, cW + dV) = \text{?}\)
\(cov(aX + bY, cW + dV) = ac \cdot cov(X, W) + ad \cdot cov(X, V) + bc\cdot cov(Y, W) + bd\cdot cov(Y, V)\)
[Statistics]
\(cov(X, Y) = \text{?}\)
\(cov(X, Y)=E[XY]-E[X]E[Y]\)
