Example:Joint probability distribution

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Example

Define a probability space, $(S,\Omega,\mathbb{P})$ as follows:

• Let $S:\eq\{0,1,2,3\}\subseteq\mathbb{N}$,
• let $\Omega:\eq$$\sigma(S)$, and
• let $\mathbb{P}:\Omega\rightarrow\mathbb{R}$ be uniform over $S$, that is to say: $\mathbb{P}:R\mapsto\frac{\vert R\vert}{\vert S\vert}$
  • specifically in this case: $\mathbb{P}(R):\eq\frac{1}{4}\vert R\vert$.

$$\xymatrix{\mathbf{X} & \mathbf{S} & \mathbf{Y} }$$ $$\xymatrix{ A & 0 \ar[l] \ar[r] & C \\ & 1 \ar[ul] \ar[dr] & \\ B & 2 \ar[l] \ar[r] & D\\ & 3 \ar[ur] \ar[ul] }$$
Experiment mappings

We define two random variables:

• $X:S\rightarrow\{A,B\}$
  • By $X(0):\eq A$, $X(1):\eq A$, $X(2):\eq B$ and $X(3):\eq B$
    • $X:0,1\mapsto A$ and $X:2,3\mapsto B$
      TODO: Is this better?
  • Think of $X$ as categorising the elementary events as "low" ($0$ and $1$) and "high" ($2$ and $3$)
• $Y:S\rightarrow\{C,D\}$^Again:
  • By $Y:0\mapsto C$ and $Y:1,2,3\mapsto D$
  • Think of $Y$ as an "is zero" measurement

Note: Our experiment here has 2 measurements, $X$ is "high or low" and $Y$ is "zero or non zero"

PAGE NOTES

We get the following:

• Events: $A\eq\{0,1\}$ (giving $\mathbb{P}[A]\eq\frac{2}{4}\eq\frac{1}{2}$), $B\eq\{2,3\}$ (giving $\mathbb{P}[B]\eq\frac{2}{4}\eq\frac{1}{2}$), $C\eq\{0\}$ (giving $\mathbb{P}[C]\eq\frac{1}{4}$) and $D\eq\{1,2,3\}$ (giving $\mathbb{P}[D]\eq\frac{3}{4}$) as events in $\Omega$, thus:
  • $A\cap C$$\eq\{0\}$ so $\mathbb{P}[A\wedge B]\eq\frac{1}{4}$
  • $A\cap D\eq\{1\}$ so $\mathbb{P}[A\wedge D]\eq\frac{1}{4}$
  • $B\cap C\eq\{\}\eq\emptyset$ so $\mathbb{P}[B\wedge C]\eq 0$
  • $B\cap D\eq\{2,3\}$ so $\mathbb{P}[B\wedge D]\eq\frac{1}{2}$
• Notice as well that $\mathbb{P}[A\cup B]\eq\mathbb{P}[A]+\mathbb{P}[B]$$\eq 1$ and $\mathbb{P}[C\cup D]\eq\mathbb{P}[C]+\mathbb{P}[D]\eq 1$ for the same reasoning

Giving the following table:

	A↓	B↓	$\mathbf{\Sigma}$
C →	$\frac{1}{4}$	$0$	$\frac{1}{4}$
D →	$\frac{1}{4}$	$\frac{2}{4}\eq\frac{1}{2}$	$\frac{3}{4}$
$\mathbf{\Sigma}$	$\frac{1}{2}$	$\frac{1}{2}$	$1$

But there are others, eg the following is if (or for) independent $X$ and $Y$ where the individual distributions of $X$ and $Y$ are the same, ie $P[X\eq A]\eq\frac{1}{2}$ still for example

	A↓	B↓	$\mathbf{\Sigma}$
C →	$\frac{1}{4}\times\frac{1}{2}\eq\frac{1}{8}$	$\frac{1}{4}\times\frac{1}{2}\eq\frac{1}{8}$	$\frac{2}{8}\eq\frac{1}{4}$
D →	$\frac{3}{4}\times\frac{1}{2}\eq\frac{3}{8}$	$\frac{3}{4}\times\frac{1}{2}\eq\frac{3}{8}$	$\frac{6}{8}\eq\frac{3}{4}$
$\mathbf{\Sigma}$	$\frac{4}{8}\eq\frac{1}{2}$	$\frac{4}{8}\eq\frac{1}{2}$	$1$

In general:

	A↓	B↓	$\mathbf{\Sigma}$
C →	$\alpha:\eq\mathbb{P}[A\cap C]$	$\beta:\eq\mathbb{P}[B\cap C]$	$\alpha+\beta\eq\mathbb{P}[C]$
D →	$\gamma:\eq\mathbb{P}[A\cap D]$	$\delta:\eq\mathbb{P}[B\cap D]$	$\gamma+\delta\eq\mathbb{P}[D]$
$\mathbf{\Sigma}$	$\alpha+\gamma\eq\mathbb{P}[A]$	$\beta+\delta\eq\mathbb{P}[B]$	$1$

Solving the equations that arise and parameterising $\alpha$ as $t$ we get:

	A↓	B↓	$\mathbf{\Sigma}$
C →	$\frac{t}{4}$	$\frac{1-t}{4}$	$\frac{1}{4}$
D →	$\frac{2-t}{4}$	$\frac{1+t}{4}$	$\frac{3}{4}$
$\mathbf{\Sigma}$	$\frac{1}{2}$	$\frac{1}{2}$	$1$

By using the constraints we see $t\in[0,1]$ produces a "valid something" (see the

TODO: XXX note just below

)

This is our case for $t\eq 1$, for $t\eq\frac{1}{2}$ it is the independent example

We have essentially parameterised all

TODO: of something, note that there are only 24 =(4C2)x(4C1) such RVs, would $t\eq\frac{\sqrt{2} }{2}$ make sense?

So what have we parameterised I wonder?

Final thoughts

Thinking about it.... really only $t\eq 0$ and $t\eq 1$ are valid values as we must deal in a whole-number of quarters Thus there are only 2 "joint distributions" for any such set up - if you think about it, $Y$ always maps 1 element of $S$ to either $C$ or $D$ and the remaining elements of $S$ to the other one, as $X$ maps 2 of $S$ to one and the other two to its other value, one of these will always never be from the single $Y$ one - so out of the 24 distributions of $(X,Y)$ - there are actually only 2 "situations" (experiments?) described by them!