Editing Marginal distribution (section)

== Definition ==
=== Marginal probability mass function ===
Given a known [[joint distribution]] of two '''discrete''' [[random variable]]s, say, {{mvar|X}} and {{mvar|Y}}, the marginal distribution of either variable – {{mvar|X}} for example – is the [[probability distribution]] of {{mvar|X}} when the values of {{mvar|Y}} are not taken into consideration. This can be calculated by summing the [[joint probability]] distribution over all values of {{mvar|Y}}. Naturally, the converse is also true: the marginal distribution can be obtained for {{mvar|Y}} by summing over the separate values of {{mvar|X}}.

:<math>p_X(x_i)=\sum_{j}p(x_i,y_j)</math>, and <math> p_Y(y_j)=\sum_{i}p(x_i,y_j)</math>

{| class="wikitable" style="text-align: center; margin: 1em auto;"
! {{diagonal split header|''Y''|''X''}} || ''x''<sub>1</sub> || ''x''<sub>2</sub> || ''x''<sub>3</sub> || ''x''<sub>4</sub> || ''p<sub>Y</sub>''(''y'') ↓
|-
! ''y''<sub>1</sub>
| {{sfrac|4|32}}  || {{sfrac|2|32}} || {{sfrac|1|32}} || {{sfrac|1|32}}
! {{sfrac|8|32}}
|-
! ''y''<sub>2</sub>
| {{sfrac|3|32}} || {{sfrac|6|32}} || {{sfrac|3|32}} || {{sfrac|3|32}}
! {{sfrac|15|32}}
|-
! ''y''<sub>3</sub>
| {{sfrac|9|32}} || 0 || 0 || 0
! {{sfrac|9|32}}
|-
! ''p<sub>X</sub>''(''x'') →
! {{sfrac|16|32}} || {{sfrac|8|32}} || {{sfrac|4|32}} || {{sfrac|4|32}}
! {{sfrac|32|32}}
|-
|+ style="caption-side: bottom;" |{{nobold|Joint and marginal distributions of a pair of discrete random variables, ''X'' and ''Y'', dependent, thus having nonzero [[mutual information]] {{math|''I''(''X''; ''Y'')}}. The values of the joint distribution are in the 3×4 rectangle; the values of the marginal distributions are along the right and bottom margins.}}
|}
A '''marginal probability''' can always be written as an [[expected value]]:
<math display="block">p_X(x) = \int_y p_{X \mid Y}(x \mid y) \, p_Y(y) \, \mathrm{d}y = \operatorname{E}_{Y} [p_{X \mid Y}(x \mid Y)]\;.</math>

Intuitively, the marginal probability of ''X''  is computed by examining the conditional probability of ''X'' given a particular value of ''Y'', and then averaging this conditional probability over the distribution of all values of ''Y''.

This follows from the definition of [[expected value]] (after applying the [[law of the unconscious statistician]])
<math display="block">\operatorname{E}_Y [f(Y)] = \int_y f(y) p_Y(y) \, \mathrm{d}y.</math>

Therefore, marginalization provides the rule for the transformation of the probability distribution of a random variable ''Y'' and another random variable {{math|''X''{{=}}''g''(''Y'')}}:
<math display="block">p_X(x) = \int_y p_{X \mid Y}(x \mid y) \, p_Y(y) \, \mathrm{d}y = \int_y \delta\big(x - g(y)\big) \, p_Y(y) \, \mathrm{d}y.</math>

=== Marginal probability density function ===
Given two '''continuous''' [[random variable]]s ''X'' and ''Y'' whose [[joint distribution]] is known, then the marginal [[probability density function]] can be obtained by integrating the [[joint probability]] density, {{mvar|f}}, over ''Y,'' and vice versa. That is

:<math>f_X(x) = \int_{c}^{d} f(x,y) \, dy  </math>   
:<math>f_Y(y) = \int_{a}^{b} f(x,y) \, dx  </math>

where <math>x\in[a,b]</math>, and <math>y\in[c,d]</math>.

=== Marginal cumulative distribution function ===
Finding the marginal [[cumulative distribution function]] from the joint cumulative distribution function is easy. Recall that:

* For '''discrete [[random variable]]s''', <math display="block">F(x,y) = P(X\leq x, Y\leq y)</math>
* For '''continuous random variables''', <math display="block">F(x,y) = \int_{a}^{x} \int_{c}^{y} f(x',y') \, dy' dx'</math>

If ''X'' and ''Y'' jointly take values on [''a'', ''b''] × [''c'', ''d''] then

:<math>F_X(x)=F(x,d)</math> and <math>F_Y(y)=F(b,y)</math>

If ''d'' is ∞, then this becomes a limit <math display="inline">F_X(x) = \lim_{y \to \infty} F(x,y)</math>. Likewise for <math>F_Y(y)</math>.