<p>In the below figure, CDF <span class="math-container">$F(\cdot)$</span> is first-order stochastically dominated by <span class="math-container">$G(\cdot)$</span>. But <span class="math-container">$X_1$</span> and <span class="math-container">$X_2$</span> fall within the support of both distributions. So it would be possible to draw <span class="math-container">$X_1$</span> from <span class="math-container">$F$</span> and <span class="math-container">$X_2$</span> from <span class="math-container">$G$</span>, <em>or</em> to draw <span class="math-container">$X_2$</span> from <span class="math-container">$F$</span> and <span class="math-container">$X_1$</span> from <span class="math-container">$G$</span>.</p>

<p>More generally, if <span class="math-container">$X_G$</span> is a draw from <span class="math-container">$G$</span> and <span class="math-container">$X_F$</span> is a draw from <span class="math-container">$F$</span> then <span class="math-container">$X_F-X_G$</span> will sometimes be positive and sometimes negative. In this sense, the dominance is only stochastic: <span class="math-container">$G$</span> produces larger draws than <span class="math-container">$F$</span> on average, but not all of the time.</p>

<p><a href="https://i.sstatic.net/t18JH.png" rel="noreferrer"><img src="https://i.sstatic.net/t18JH.png" alt="enter image description here"></a></p>

CC BY-SA 4.0
2018-10-26T11:24:09.863Z
Great  explanation!
High GPA
CC BY-SA 4.0
2018-10-26T11:23:22.880Z
2018-10-26T11:23:22.880Z
Ubiquitous
<p>I think the CDF is pretty much fixed, so the FOSD (first order stochastic dominance) is pretty much non-stochastic. Why does it have a "stochastic" in its name?</p>

<financial-economics><decision-theory><portfolio-theory>
Why stochastic dominance is "stochastic"?
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# Why stochastic dominance is "stochastic"?

(Asked by: username_0 on 2018-10-26)

I think the CDF is pretty much fixed, so the FOSD (first order stochastic dominance) is pretty much non-stochastic. Why does it have a "stochastic" in its name?

**Tags:** financial-economics, decision-theory, portfolio-theory


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## 1 answers

### Answer by username_1 on 2018-10-26. Score: 11 | Accepted answer

In the below figure, CDF $F(\cdot)$ is first-order stochastically dominated by $G(\cdot)$. But $X\_1$ and $X\_2$ fall within the support of both distributions. So it would be possible to draw $X\_1$ from $F$ and $X\_2$ from $G$, *or* to draw $X\_2$ from $F$ and $X\_1$ from $G$.

More generally, if $X\_G$ is a draw from $G$ and $X\_F$ is a draw from $F$ then $X\_F-X\_G$ will sometimes be positive and sometimes negative. In this sense, the dominance is only stochastic: $G$ produces larger draws than $F$ on average, but not all of the time.

[![enter image description here](https://i.sstatic.net/t18JH.png)](https://i.sstatic.net/t18JH.png)

> **Comments:**
> **username_0**: Great  explanation!  

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