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p(x_1,...,x_{k-1}) = \frac{1}{B(\alpha) } \left[ \prod^{K}_{i=1} x_i^{\alpha_i - 1} \right] E[X_1] = \frac{1}{B(\alpha) } \int^1_0 ...\int^{1 - \sum^{K}_{i=2}x_i}_0 x_1 \left[\prod^{K}_{i=1} x_i^{\alpha_i - 1} \right] d x_1 ... d x_{k}

For a Dirichlet variable, I know the means and covariances, that is, E[X_i] = \alpha_i/\alpha_0 Cov[X_i,X_j] = \frac{ \alpha_i (\alpha_0I[i=j] - \alpha_j)}{\alpha_0^2(\alpha_0 + 1)} But how can I prove these facts?

Hello all, Does anyone know where I could find a formal proof that \frac{\text{MS between}}{\text{MS within}} has a F distribution under the null in ANOVA?

Homework Statement W(\theta - \delta) the loss function. \theta the true parameter. \delta an estimator of \theta W a smooth, non-negative, symmetric, convex function. p(\theta | x) the posterior density of the parameter \theta. Prove that, for normal posterior density p(\theta | x)...

But what if the f_{n} are defined over sets of measure non-zero, but that the sum of the measure of those sets converges to zero?

Is the integral over a set of measure zero always equals to zero? Can the integral be undefined?

Homework Statement Let (R, \mathcal{B}, \mu_F) be a measure space, where \mathcal{B} is the Borel \sigma-filed and \mu_F is the Lebesgue-Stieljes measure generated from F(x) = \sum^\infty_{n=1}2^{-n}I(x \ge n^{-1}) + (e^{-1} - e^{-x})I(x \ge 1) Use the uniqueness of measure extension in...