SRaI
@LMU
@LMU
125
0.0 (0)
Kartei Details
| Karten | 125 |
|---|---|
| Sprache | English |
| Kategorie | Informatik |
| Stufe | Universität |
| Erstellt / Aktualisiert | 04.10.2019 / 11.10.2019 |
| Lizenzierung | Keine Angabe |
| Weblink |
https://card2brain.ch/box/20191004_srai
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| Einbinden |
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Name bayes roule
\(P(A|B) = \frac{P(B|A) ~\cdot~ P(A)}{P(B)}\)
What is a random variable?
- random variable y maps from event-space omega to real values
Define the expected value and variance
\(E(Y) = \int_{-\infty}^{\infty}u ~f(u)~du = \mu\)
\(Var(Y) = \int_{-\infty}^{\infty}(y - \mu)^2~f(y)~dy = E((Y - \mu)^2) = \sigma^2 = E(Y^2) - \mu^2\)
Define the exponential family
\(f(y,\theta) = exp(t^T(y)~\theta - K(\theta)~h(y))\)
- with t(y) = statistics = function of data
- theta = parameter (vector)
- K(theta) = normalisation constant s.t. integral (f(y,theta)) = 1
- h(y) >= 0, unimportant
- and \(\frac{\partial K(\theta)}{\partial\theta} = E(t(Y))\)
What is the t-distribution good for?
- statistical test for mean of normal distributed variables
- when variance is unknown (estimated from data)
Define covariance for Y1, Y2.
What about independence? What does this imply?
\(Cov(Y_1, Y_2) = E((Y_1 - E(Y_1)(Y_2 - E(Y_2))) = E(Y_1~Y_2)-E(Y_1)E(Y_2)\)
Cov(Yj, Yk) = 0 if Yj, Yk are independent
- f(yj, yk) = f(yj) * f(yk)
- E(Yj, Yk) ) E(Yj) * E(Yk)
Define correlation.
- Corr(Yj, Yk) = Cov(Yj, Yk) / sqrt(Var(Yj) * Var(Yk))
Name iterated expectation.
\(E(Y) = E_X(E(Y|X))\\Var(Y) = E_X(Var(Y|X) + Var_X(E(Y|X))\)
