View source for Notes:Probability of an RV being less than another

{{ProbMacros}}
::: {{Green highlight|This was solved/worked out, see: [[Probability of i.i.d random variables being in an order and not greater than something]]}}
__TOC__
==Solution==
* {{MM|\P{X\le Y\le Z}\eq\sum_z\sum_{y\le z}\sum_{x\le y}\P{X\eq x\cap Y\eq y\cap Z\eq z} }} - ''duh!'' - silly me! 
** Integral form is obvious.
=SILLY STUFF=
==Notes==
Here I investigate:
# {{M|\P{X\le Y} }}, and
# {{M|\P{X\le Y\le Z} }} which of course is short for {{M|\P{\big(X\le Y\big)\cap\big(Y\le Z\big)} }}
==Case 1:==
* {{MM|\P{X\le Y}\eq\sum_{y\in S_Y}\Big( \P{Y\eq y}\times\Pcond{X\le Y}{Y\eq y}\Big) }}
*: {{MM|\eq\sum_{y\in S_y}\Big(\P{Y\eq y}\times\P{X\le y} \Big)}}
** So: {{MM|\P{X\le Y}\eq\sum_{y\in S_Y}\Big( f_Y(y)\times F_X(y)\Big) }} where {{M|f_Y}} is the [[probability mass function]] of {{M|Y}} and {{M|F_X}} is the [[cumulative probability function]] of {{M|X}}
*** Note that there are bounds on {{M|X}} hiding in here, as another way to write this is:
**** {{MM|\P{X\le Y}\eq\sum_{y\in S_y}\left( f_Y(y)\times \left[\sum_{x\in S_X,\ x\le y}\P{X\eq x}\right]\right)}}
** The "integral form" is obviously:
*** {{MM|\P{X\le Y}\eq\int_{S_y}\Big(f_Y(y)\!\ F_X(y)\Big)\mathrm{d}y}} from the infinitesimal-style abuse of notation: {{M|\P{Y\eq y}\mathrm{d}y\cdot \P{X\le y} }}
**** Which may be written as:
***** {{MM|\P{X\le Y}\eq\int_{S_y}\left(f_Y(y)\cdot\left[\int_{x\in S_x,\ x\le y}f_X(x)\right]\mathrm{d}x\right)\mathrm{d}y}}
==Case 2:==
* {{MM|\P{X\le Y\le Z}\eq\sum_{z\in S_Z}\P{Z\eq z}\cdot\Pcond{X\le Y\le Z}{Z\eq z} }}
*: {{MM|\eq\sum_{z\in S_Z}\P{Z\eq z}\cdot\Pcond{X\le Y}{Y\le z}  }}
*: {{MM|\eq\sum_{z\in S_Z}\left(\P{Z\eq z}\cdot\frac{\P{X\le Y\le z} }{\P{Y\le z} }\right)}}
*: {{MM|\eq\sum_{z\in S_Z}\left(\frac{\P{Z\eq z} }{\P{Y\le z} }\left[\sum_{y\in S_Y,\ y\le z}\P{Y\eq y}\cdot\Pcond{X\le Y}{Y\eq y}\right]\right)}}
*: {{MM|\eq\sum_{z\in S_Z}\left(\frac{\P{Z\eq z} }{\P{Y\le z} }\sum_{y\in S_Y,\ y\le z} \P{Y\eq y}\cdot\P{X\le y} \right) }}
Integral form... coming soon!