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The legend is wrong (or very misleading), while the code snippet displays the correct output. In the third example, given $E$, the highlighted node in grey, $A$ and $S$ are not d-separated. In this elementary configuration, $E$ is called a collider, and in a collider, conditioning on the common effect $E$ makes $A$ and $S$ dependent on each other. See for ...

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That is just a function of the question being asked here. You want to know if you can say Travel is dependent on Education to a degree not explained by the Occupation and Residence factors. You're not asking what Education is dependent on, nor are you asking whether Age and Sex are or are not a factors in Travel. You either know or assume the conditional ...

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The staging does explicitly state anything about the statistical relation of the variables, but rather tells us which conditional distributions we care about. For example, suppose some variable A is a first stage variable and B is a second stage variable. Then, the two probability distributions we care about are $P(A)$ and $P(B | A)$. This is because at the ...

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The generator matrix is given by $$G = \begin{pmatrix} -(\lambda_1+\lambda_2)&\lambda_2&\lambda_1&0\\ \mu_2& -(\lambda_1+\mu_2)& 0 & \lambda_1\\ \mu_1& 0 & -(\lambda_2+\mu_1)& \lambda_2\\ 0 & \mu_1 & \mu_2 & -(\mu_1+\mu_2), \end{pmatrix}$$ and the transition matrix for the embedded Markov chain is given by  ...

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When the $t$-tests are performed, they assume that the other variables are already in the model. For example, suppose you were building a model where the dependent variable was the weight of a book, and the independent variables were $x_2$ (the number of pages in the book) and $x_3$ (the thickness of the book). If you fit a model with both of these ...

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P{A(B+C)}=P(AB+BC)=P(AB)+P(AC)-P(ABC) =P(A)P(B)+P(A)P(C)-P(A)P(BC) [A,B,C are mutually independent] =P(A)[P(B)+P(C)-P(BC)] =P(A)P(B+C) Hence A and B+C are independent.

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