# The distribution for one of the components of a minimized sum?

I have a complicated thing I would like to find the distribution for.

Let's say I have a random variable $X\sim F_X(x)$ supported over $(0,1)$. I have two independent draws from $F_X$, which are $x_1$ and $x_2$.

Similarly, I have a random variable $Y\sim F_Y(y)$ supported over $(0,1)$. I have two independent draws from $F_Y$, which are $y_1$ and $y_2$.

I am not even sure what words to use to express this properly, but I want to find the distribution for (or any way to express) the "$x$" part of the $Minumum[x_1 + y_1, x_2 +y_2]$. That is probably not clear enough so let me try to explain more:

If $x_1 + y_1 < x_2 +y_2$ I want some way of expressing a distribution for $x_1$ that is more specific than $F_X$ since now we have more information about it. (And if $x_1 + y_1 > x_2 +y_2$ the of course I would want the way to describe $x_2$).

Does this question make sense? And if so, can anyone help me, even if it's just to have better terminology for describing what I'm looking for?

Thanks so much!

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Certainly understandable. I would be more comfortable if capital letters were used. Have not thought it through, but it seems very reasonable to think that the distribution of the $Y_i$ is irrelevant. – André Nicolas May 21 '12 at 2:01
Thanks Andre! I could be wrong, but I think the distribution of Y is relevant. For example, if Y is always the same constant, then y1=y2 and the distribution I am looking for is just the minimum of two draws from $F_X$ , which is $1-(1-x)^2$. But, if Y can vary a lot, then it is possible that the $X_i$ I am looking for is not the minimum of $X_i$ and $X_j$ – Angada May 21 '12 at 3:14
I am not saying it is the minimum, only that it has the same distribution as the minimum. But need to verify, interesting question. – André Nicolas May 21 '12 at 4:30
Andre: what is the guideline for when to use capitalized vs. lowercase variables? I have seen both, but I don't know when to use which. Thanks for your help! – Angada May 21 '12 at 17:23
Habits differ. In teaching probability, I found that students are least confused if they always use caps for random variables. The issue comes up most commonly in random sampling. It is tempting to think of a sample of $10$ as $10$ numbers. But that is very unhelpful if we then ask for the probability that the sample mean differs from the true mean by less than $k$. – André Nicolas May 21 '12 at 17:30

I might write it this way. Let $T = 1$ when $X_1 + Y_1 < X_2 + Y_2$, $2$ when $X_1 + Y_1 \ge X_2 + Y_2$. You want the distribution of $X_T$.

Now $P(X_T < x|Y_1, Y_2) = P(X_1 < x, X_1 +Y_1 < X_2 + Y_2 )) + P(X_2 < x, X_1 + Y_1 \le X_2 + Y_2)$. Suppose $X_1, X_2, Y_1, Y_2$ are all independent with continuous distributions (so I don't have to distinguish between $<$ and $\le$), and $|Y_2 - Y_1| = V$. There are several different cases, depending on the ordering of $x$, $V$ and $1-V$. For example, if $x < \min(V, 1-V)$, $P(X_T < x | Y_1, Y_2)$ is obtained by integrating $f_X(x_1) f_X(x_2)$ over a region that looks like this:

Then you'll have to integrate the result times $f_Y(y_1) f_Y(y_2)$ over the unit square to get unconditional probability.

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The distribution of $X_1$ conditional on the event $[X_1+Y_1\lt X_2+Y_2]$ has density $g$ defined by $$g(x)=\frac1cf_X(x)u(x),$$ with $$u(x)=\mathrm P(x+Y_1\lt X_2+Y_2), \qquad c=\int_{-\infty}^{+\infty} f_X(z)u(z)\mathrm dz.$$ Since $u$ is nonincreasing, $g$ puts more weight than $f_X$ on the small values.

Note that the distribution of $\min\{X_1,X_2\}$ has density $m$, where $$m(x)=2f_X(x)v(x),\quad v(x)=\mathrm P(x\lt X_2)=1-F_X(x).$$

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Thank you. I am not sure how to interpret $c^{-1}$ though. Is it the inverse of the indefinite integral of $f_X(z)u(z)dz$? Then wouldn't I have $z$'s in the $g(x)$ equation? Or, in $g(x)$, do I sub in $f_x(x)u(x)$ as in $c^{-1}(f_x(x)u(x))$? I am sorry I am not understanding. – Angada May 21 '12 at 17:20
Also, is it easier or equivalent to stay in terms of CDFs rather than densities? I know I can change back to CDFs at the end, but since all I want are the CDFs, so I don't need the PDFs unless they are necessary to calculate what I'm looking for. – Angada May 21 '12 at 17:22