Joint Probability Distributions - Revision history

Rb: /* Independence */

2014-09-10T13:04:21Z

‎Independence

2014-09-10T13:03:30Z

‎Independence

2014-09-10T13:02:27Z

‎Independence

2014-09-10T12:59:34Z

‎Marginal Probabilities

2013-10-01T14:56:22Z

‎Covariance

2013-08-19T14:06:53Z

‎Example 1

2013-08-19T14:04:40Z

‎Examples

2013-08-19T14:02:18Z

‎Example 1

2013-08-19T13:59:42Z

2013-08-19T08:25:03Z

‎Covariance

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 <p><math>p\left( x,y\right) =p_{X}\left( x\right) p_{Y}\left( y\right) \;\;\;\;\; \text{for all }x,y.</math></p>
 <p>Each joint probability is the product of the corresponding marginal probabilities. Independence also means that <math>\Pr \left( Y=y\right) </math> would not be affected by knowing that <math>X=x:</math> knowing the value taken on by one random variable does not affect the probabilities of the outcomes of the other random variable. This is also expressed by the following relationship that ''is only valid if X and Y are independent'':</p>
-<p><math>p\left( x|y\right) = frac{p\left( x,y\right)}{p(y)}.</math></p>
+<p><math>p\left( x|y\right) = \frac{p\left( x,y\right)}{p(y)}.</math></p>
 <p>Another corollary of this is that if two random variables <math>X</math> and <math>Y</math> are independent, then there can be no relationship of any kind, linear or non-linear, between them.</p></li>
 <li><p>The joint probabilities and marginal probabilities (taken from Example 1 above) are</p>

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 <ul>
 <li><p><math>X</math> and <math>Y</math> are '''independent''' random variables if and only if</p>
-<p><math>p\left( x,y\right) =p_{X}\left( x\right) p_{Y}\left( y\right) \;\;\;\;\; \text{for \textbf{all }}x,y.</math></p>
+<p><math>p\left( x,y\right) =p_{X}\left( x\right) p_{Y}\left( y\right) \;\;\;\;\; \text{for all }x,y.</math></p>
 <p>Each joint probability is the product of the corresponding marginal probabilities. Independence also means that <math>\Pr \left( Y=y\right) </math> would not be affected by knowing that <math>X=x:</math> knowing the value taken on by one random variable does not affect the probabilities of the outcomes of the other random variable. This is also expressed by the following relationship that ''is only valid if X and Y are independent'':</p>
 <p><math>p\left( x|y\right) = frac{p\left( x,y\right)}{p(y)}.</math></p>

@@ Line 499: / Line 499: @@
 <p><math>p\left( x|y\right) = frac{p\left( x,y\right)}{p(y)}.</math></p>
 <p>Another corollary of this is that if two random variables <math>X</math> and <math>Y</math> are independent, then there can be no relationship of any kind, linear or non-linear, between them.</p></li>
-<li><p>The joint probabilities and marginal probabilities for Example 1 in section [jtprobex] are</p>
+<li><p>The joint probabilities and marginal probabilities (taken from Example 1 above) are</p>
 <table border="1">

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 <math>p_{X}\left( x\right) =\sum_{y}p\left( x,y\right) ,\;\;\;\;\;p_{Y}\left(y\right) =\sum_{x}p\left( x,y\right) .</math>
-This can be illustrated using Example 1 of Section [jtprobex] again:
+This can be illustrated using Example 1 from above:
 <math>\begin{aligned}

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 &=&12000-\left( 90\right) \left( 130\right) \\
 &=&300.\end{aligned}</math>
 === Strength of Association and Units of Measurement ===

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 === Example 1 ===
-{{Note|This workaround may harm accessibility.}}
 Let <math>H</math> and <math>W</math> be the random variables representing the population of weekly incomes of husbands and wives, respectively, in some country. There are only three possible weekly incomes, £0, £100 or £ 200. The joint probability distribution of <math>H</math> and <math>W</math> is represented as a table:

← Older revision		Revision as of 14:04, 19 August 2013
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	=== Example 1 ===		=== Example 1 ===
		+
		+	{{Note\|This workaround may harm accessibility.}}

	Let <math>H</math> and <math>W</math> be the random variables representing the population of weekly incomes of husbands and wives, respectively, in some country. There are only three possible weekly incomes, £0, £100 or £ 200. The joint probability distribution of <math>H</math> and <math>W</math> is represented as a table:		Let <math>H</math> and <math>W</math> be the random variables representing the population of weekly incomes of husbands and wives, respectively, in some country. There are only three possible weekly incomes, £0, £100 or £ 200. The joint probability distribution of <math>H</math> and <math>W</math> is represented as a table:

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 === Example 1 ===
 Let <math>H</math> and <math>W</math> be the random variables representing the population of weekly incomes of husbands and wives, respectively, in some country. There are only three possible weekly incomes, £0, £100 or £ 200. The joint probability distribution of <math>H</math> and <math>W</math> is represented as a table:
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 or that in this population, <math>5\%</math> of husbands and wives have each a zero weekly income.
 In this example, the nature of the experiment underlying the population data is not explicitly stated. However, in the next example, the experiment is described, the random variables defined in relation to the experiment, and their probability distribution deduced directly.

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 </table>
-Recall that a value of <math>1</math> represented <math>£\100</math> (and similar <math>2=£\200</math>). Therefore, using the same strategy of multiplication within cells, and adding up along each row in turn, we find the following
+Recall that a value of <math>1</math> represented <math>£ 100</math> (and similar <math>2=£ 200</math>). Therefore, using the same strategy of multiplication within cells, and adding up along each row in turn, we find the following
 <math>\begin{aligned}