Revision as of 00:12, 2 December 2017 (view source) Alec (Talk | contribs) (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)...")		Revision as of 01:11, 2 December 2017 (view source) Alec (Talk | contribs) m Newer edit →
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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:

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Revision as of 01:11, 2 December 2017

$\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} }$

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:

1. $\P{X\le Y}$, and
2. $\P{X\le Y\le Z}$ 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:
      • $$\P{X\le Y}\eq\sum_{y\in S_y}\left( f_Y(y)\times \left[\sum_{x\in S_X,\ x\le y}\P{X\eq x}\right]\right)$$
  • 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:
        • $$\P{X\le Y}\eq\int_{S_y}\left(f_Y(y)\cdot\left[\int_{x\in S_x,\ x\le y}f_X(x)\right]\mathrm{d}x\right)\mathrm{d}y$$

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!