Editing Marginal distribution (section)

== Marginal distribution vs. conditional distribution ==

=== Definition ===
The '''marginal probability''' is the probability of a single event occurring, independent of other events. A '''[[Conditional probability distribution|conditional probability]]''', on the other hand, is the probability that an event occurs given that another specific event ''has already'' occurred. This means that the calculation for one variable is dependent on another variable.<ref>{{Cite web|url=https://study.com/academy/lesson/marginal-conditional-probability-distributions-definition-examples.html|title=Marginal & Conditional Probability Distributions: Definition & Examples|website=Study.com|language=en|access-date=2019-11-16}}</ref>

The conditional distribution of a variable given another variable is the joint distribution of both variables divided by the marginal distribution of the other variable.<ref>{{Cite web|url=https://www.math.fsu.edu/~paris/Pexam/|title=Exam P [FSU Math]|website=www.math.fsu.edu|access-date=2019-11-16}}</ref> That is,

* For '''discrete [[random variable]]s''',<math display="block">p_{Y|X}(y|x) = P(Y=y \mid X=x) = \frac{P(X=x,Y=y)}{P_X(x)}</math>
* For '''continuous random variables''',<math display="block">f_{Y|X}(y|x)=\frac{f_{X,Y}(x,y)}{f_X(x)}</math>

=== Example ===
Suppose there is data from a classroom of 200 students on the amount of time studied (''X'') and the percentage of correct answers (''Y'').<ref>{{Citation|title=Marginal and conditional distributions|url=https://www.khanacademy.org/math/ap-statistics/analyzing-categorical-ap/distributions-two-way-tables/v/marginal-distribution-and-conditional-distribution|language=en|access-date=2019-11-16}}</ref> Assuming that ''X'' and ''Y'' are discrete random variables, the joint distribution of ''X'' and ''Y'' can be described by listing all the possible values of ''p''(''x<sub>i</sub>'',''y<sub>j</sub>''), as shown in Table.3.
{| class="wikitable" style="text-align: center; width:560px; margin: 1em auto;"
! {{diagonal split header|''Y''|''X''}} 
! colspan="6" |Time studied (minutes)
|-
| rowspan="7" {{vertical header|cellstyle=background-color:#ececec|'''% correct'''}}
! 
! scope="col" |''x''<sub>1</sub> (0-20)
! scope="col" |''x''<sub>2</sub> (21-40)
!''x''<sub>3</sub> (41-60)
!''x''<sub>4</sub>(>60)
!''p''<sub>''Y''</sub>(''y'') ↓
|-
|{{rh|align=center}} |''y''<sub>1</sub> (0-20)
|{{sfrac|2|200}}
|0
|0
|{{sfrac|8|200}}
|{{rh|align=center}} |{{sfrac|10|200}}
|-
|{{rh|align=center}} |''y''<sub>2</sub> (21-40)
|{{sfrac|10|200}}
|{{sfrac|2|200}}
|{{sfrac|8|200}}
|0
|{{rh|align=center}} |{{sfrac|20|200}}
|-
|{{rh|align=center}} |''y''<sub>3</sub> (41-59)
|{{sfrac|2|200}}
|{{sfrac|4|200}}
|{{sfrac|32|200}}
|{{sfrac|32|200}}
|{{rh|align=center}} |{{sfrac|70|200}}
|-
|{{rh|align=center}} |''y''<sub>4</sub> (60-79)
|0
|{{sfrac|20|200}}
|{{sfrac|30|200}}
|{{sfrac|10|200}}
|{{rh|align=center}} |{{sfrac|60|200}}
|-
|{{rh|align=center}} |''y''<sub>5</sub> (80-100)
| align="center" | 0
| align="center" | {{sfrac|4|200}}
| align="center" |{{sfrac|16|200}}
| align="center" |{{sfrac|20|200}}
| {{rh|align=center}} |{{sfrac|40|200}}
|-
|{{rh|align=center}} |''p<sub>X</sub>''(''x'') →
! align="center" |{{sfrac|14|200}}
! align="center" |{{sfrac|30|200}}
! align="center" |{{sfrac|86|200}}
! align="center" |{{sfrac|70|200}}
! align="center" |1
|-
|+ style="caption-side: bottom;" |{{nobold|[[Two-way table]] of dataset of the relationship in a classroom of 200 students between the amount of time studied and the percentage correct}}
|}
The '''marginal distribution''' can be used to determine how many students scored 20 or below: <small><math>p_Y(y_1) = P_Y(Y=y_1) = \sum_{i=1}^4 P(x_i,y_1) = \frac{2}{200} + \frac{8}{200} = \frac{10}{200}</math></small>, meaning 10 students or 5%.

The '''[[conditional distribution]]''' can be used to determine the probability that a student that studied 60 minutes or more obtains a scored of 20 or below: <small><math>p_{Y|X}(y_1|x_4) = P(Y=y_1|X=x_4) = \frac{P(X=x_4,Y=y_1)}{P(X=x_4)} = \frac{8/200}{70/200} = \frac{8}{70} = \frac{4}{35}</math></small>, meaning there is about a 11% probability of scoring 20 after having studied for at least 60 minutes.