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Fenster schliessen

Name bayes roule

\(P(A|B) = \frac{P(B|A) ~\cdot~ P(A)}{P(B)}\)

Fenster schliessen

What is a random variable?

  • random variable y maps from event-space omega to real values
Fenster schliessen

Define the expected value and variance

\(E(Y) = \int_{-\infty}^{\infty}u ~f(u)~du = \mu\)

\(Var(Y) = \int_{-\infty}^{\infty}(y - \mu)^2~f(y)~dy = E((Y - \mu)^2) = \sigma^2 = E(Y^2) - \mu^2\)

Fenster schliessen

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 \(\frac{\partial K(\theta)}{\partial\theta} = E(t(Y))\)
Fenster schliessen

What is the t-distribution good for?

  • statistical test for mean of normal distributed variables
  • when variance is unknown (estimated from data)
Fenster schliessen

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)
Fenster schliessen

Define correlation.

  • Corr(Yj, Yk) = Cov(Yj, Yk) / sqrt(Var(Yj) * Var(Yk))
Fenster schliessen

Name iterated expectation.

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