# Lernkarten

Karten 125 Karten 1 Lernende English Universität 04.10.2019 / 11.10.2019 Keine Angabe
0 Exakte Antworten 125 Text Antworten 0 Multiple Choice Antworten

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.

$$E(Y) = E_X(E(Y|X))\\Var(Y) = E_X(Var(Y|X) + Var_X(E(Y|X))$$