Difference between revisions of "Notes:Probability of an RV being less than another"
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(Created page with "{{ProbMacros}} __TOC__ ==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)...") |
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{{ProbMacros}} | {{ProbMacros}} | ||
__TOC__ | __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== | ==Notes== | ||
Here I investigate: | Here I investigate: | ||
Revision as of 01:11, 2 December 2017
\newcommand{\P}[2][]{\mathbb{P}#1{\left[{#2}\right]} } \newcommand{\Pcond}[3][]{\mathbb{P}#1{\left[{#2}\!\ \middle\vert\!\ {#3}\right]} } \newcommand{\Plcond}[3][]{\Pcond[#1]{#2}{#3} } \newcommand{\Prcond}[3][]{\Pcond[#1]{#2}{#3} }
\newcommand{\E}[1]{ {\mathbb{E}{\left[{#1}\right]} } } \newcommand{\Mdm}[1]{\text{Mdm}{\left({#1}\right) } } \newcommand{\Var}[1]{\text{Var}{\left({#1}\right) } } \newcommand{\ncr}[2]{ \vphantom{C}^{#1}\!C_{#2} }
Contents
[hide]Solution
- \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:
- \P{X\le Y} , and
- \P{X\le Y\le Z} which of course is short for \P{\big(X\le Y\big)\cap\big(Y\le Z\big)}
Case 1:
- \P{X\le Y}\eq\sum_{y\in S_Y}\Big( \P{Y\eq y}\times\Pcond{X\le Y}{Y\eq y}\Big)
- \eq\sum_{y\in S_y}\Big(\P{Y\eq y}\times\P{X\le y} \Big)
- So: \P{X\le Y}\eq\sum_{y\in S_Y}\Big( f_Y(y)\times F_X(y)\Big) where f_Y is the probability mass function of Y and F_X is the cumulative probability function of X
- Note that there are bounds on X hiding in here, as another way to write this is:
- \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)
- Note that there are bounds on X hiding in here, as another way to write this is:
- The "integral form" is obviously:
- \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: \P{Y\eq y}\mathrm{d}y\cdot \P{X\le y}
- Which may be written as:
- \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
- Which may be written as:
- \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: \P{Y\eq y}\mathrm{d}y\cdot \P{X\le y}
Case 2:
- \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}
- \eq\sum_{z\in S_Z}\P{Z\eq z}\cdot\Pcond{X\le Y}{Y\le z}
- \eq\sum_{z\in S_Z}\left(\P{Z\eq z}\cdot\frac{\P{X\le Y\le z} }{\P{Y\le z} }\right)
- \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)
- \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!
