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My statistics knowledge is somewhat poor, so I have to ask one (dumb) question.

Let $\beta$ be a real number in the interval $\big[0, \frac{1}{2}\big)$ and $\mathcal{D}_1, \mathcal{D}_2, \mathcal{D}_3$ be three distributions over a space $\mathcal{X}$, with the property that $\mathcal{D}_1 = \beta \cdot \mathcal{D}_2 + (1-\beta) \cdot \mathcal{D}_3$. What is the statistical variational distance between $\mathcal{D}_1$ and $\mathcal{D}_2$?

Thanks you a lot!

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closed as off-topic by Emil Jeřábek supports Monica, Jeffε, Kaveh, Jan Johannsen, Hsien-Chih Chang 張顯之 Aug 18 '17 at 6:07

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Using the relation between total variation and $L_1$/$\ell_1$ distance of the probability/distribution/mass functions, we have $$\begin{align} d_{\rm TV}(D_1, D_2) &= \frac{1}{2}\lVert D_1-D_2\rVert_1 = \frac{1}{2}\lVert \beta D_2 +(1-\beta)D_3 - D_2\rVert_1\\ &= \frac{1-\beta}{2}\lVert D_3 - D_2\rVert_1 = (1-\beta)d_{\rm TV}(D_2, D_3). \end{align}$$

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  • $\begingroup$ Is that also available for an infinite probability space? $\endgroup$ – Cip Baetu Aug 10 '17 at 4:36
  • $\begingroup$ @tutifresh Yes, it is. $\endgroup$ – Clement C. Aug 10 '17 at 11:13

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