Konstantin
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Answer: Your code seems to be in line with the cited paper, and the incosistency is an artefact of non-standard definition of principal components used in scikit-learn of versions < 0.22 (this is ...

Answer: Whatever the distribution of $X_1,...,X_n$, $$\mathbb{E} Y_2 \geq \mathbb{E} Y_1.$$ Details: For any $n$ numbers $X_1,..., X_n$ it is true that $$\sum_i |X_i| \geq |\sum_i X_i|$$ and ...

Answer: There is a mistake in the formula for $\theta$. The correct computation must align autocovariances of the MA components of two representations. The correct formula is $$\theta = \frac{\... View answer 5 votes Answer: Posterior of \sigma^2|Y_1,..., Y_n is an instance of inverse gamma distribution with the probability density$$ p(\sigma^2|Y_1,...,Y_n) = \frac{\beta^\alpha}{\Gamma(\alpha)} (\sigma^2)^{-\...

Answer: Correct, $\Sigma^{-1}(I-P)$, positive definite, would also be a kernel. Correct, $\Sigma^{-1}$, positive definite is a valid kernel. $\Sigma^{-1} K = \frac{1}{\sigma^2_1}I-\frac{\sigma^2_2}{\... View answer Accepted answer 4 votes The answer: $$P_{X_1 > X_2} = \frac{B(\frac{1}{2};a_2,a_1)}{B(a_1,a_2)},$$ where$B(\alpha,\beta) = \int_0^1 t^{\alpha-1}(1-t)^{\beta-1}dt$is the Beta function and$...

Short answer: Indeed, when the same customer may be approached at most $n$ times, it is optimal to start with offer $y_1=\frac{n-1}{n}x$ and decrease the price by $\frac{x}{n}$ with every refusal. ...

Answer: Although it does not resemble any distribution I know, it is possible to obtain a compact expression for the probability density. Denote $W=p_1+p_2$ and $Z=p_1+p_3$ then their joint density ...

The answer : \begin{align} p(y_1=1|y_1^o = 1, y_{-1}^o = \mathbf{0} ) = \frac{ p(y_1^o=1|y_1=1) \cdot \sum_{y_2=0}^1 \sum_{y_3=0}^1 \sum_{y_4=0}^1 \prod_{i=2}^4 p(y_i^o=0|y_i) \phi_{1,y_2} \phi_{...

You are on the right track. Substitute expressions for the densities of normals into the Bayes law (and don't keep track of constants !!) \begin{align} f(\theta|X) &\propto \exp(-\frac{1}{2}(X-\...

Just follow the instructions: A horizontal move represents deletion A vertical move represents insertion A diagonal move represents match 'deletion', 'insertion' and 'match' refer to the ...

Answer: Every unit-root state-space model has an equivalent ARIMA(0,1,1) representation, but not every ARIMA(0,1,1) model has an equivalent state-space model representation. The following holds true ...

Answer: A decent first pass would be a generalized ARMA model followed by the analysis of impulse response functions (IRFs). Interesting IRF measurements: peak, delay until peak, halftime of peak ...

Try numpy.random.dirichlet in Python. Once you install Python and the numpy package (to install both in the easiest way start here), you can generate your samples in a new file by the code import ...

Intuitively, quadrature approximation of an integral is replacing the function $f(x)$ under the integral by a close enough step-function $\hat f (x)$ (piecewise constant with a finite number of jumps)....
Answer: there is no way to do it, the premise is wrong. For an AR(1) process $x_t = \rho x_{t-1} + e_t$ where $e_t$ is a white-noise process, the equation $$\mathbb{V}x_t = \rho^2 \mathbb{V}x_t + \... View answer 0 votes Your source has Theorem 2', which says that Z SOSD X if for every concave function u$$ \mathbb{E}u(Z) \geq \mathbb{E}u(X), whatever the sign of the first derivative, u'. Now, \begin{align} ... View answer Accepted answer 0 votes According to the Bayes law the joint probability of a sample observation (x,y) can be decomposed into the familiar product: P(x,y) = P(x|y)P(y). In other words, we can always partition the set ...