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

CC BY-SA 3.0
2017-08-10T04:36:31.173Z
Is that also available for an infinite probability space?
Cip Baetu
CC BY-SA 3.0
2017-08-10T11:13:24.693Z
@tutifresh Yes, it is.
Clement C.
CC BY-SA 3.0
2017-08-10T01:13:39.047Z
2017-08-10T01:13:39.047Z
Clement C.
<p>My statistics knowledge is somewhat poor, so I have to ask one (dumb) question.</p>

<p>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 <a href="https://en.wikipedia.org/wiki/Total_variation_distance_of_probability_measures" rel="nofollow noreferrer">statistical variational distance</a> between $\mathcal{D}_1$ and $\mathcal{D}_2$?</p>

<p>Thanks you a lot!</p>

<st.statistics><edit-distance>
L1 / Variational Distance between distributions
Cip Baetu
# L1 / Variational Distance between distributions

(Asked by: username_0 on 2017-08-09)

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](https://en.wikipedia.org/wiki/Total_variation_distance_of_probability_measures) between $\mathcal{D}\_1$ and $\mathcal{D}\_2$?

Thanks you a lot!

**Tags:** st.statistics, edit-distance

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## 1 answers

### Answer by username_1 on 2017-08-10. Score: 5 | Accepted answer

Using the relation between total variation and $L\_1$/$\ell\_1$ distance of the probability/distribution/mass functions, we have
$$\begin{align}
d\_{\rm TV}(D\_1, D\_2) &= \frac{1}{2}\lVert D\_1-D\_2\rVert\_1
= \frac{1}{2}\lVert \beta D\_2 +(1-\beta)D\_3 - D\_2\rVert\_1\\
&= \frac{1-\beta}{2}\lVert D\_3 - D\_2\rVert\_1
= (1-\beta)d\_{\rm TV}(D\_2, D\_3).
\end{align}$$

> **Comments:**
> **username_0**: Is that also available for an infinite probability space?  
> **username_1**: @tutifresh Yes, it is.  

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