Editing Marginal distribution (section)

==Real-world example==
Suppose that the probability that a pedestrian will be hit by a car, while crossing the road at a pedestrian crossing, without paying attention to the traffic light, is to be computed. Let H be a [[discrete random variable]] taking one value from {Hit, Not&nbsp;Hit}. Let L (for traffic light) be a discrete random variable taking one value from {Red, Yellow, Green}.

Realistically, H will be dependent on L. That is, P(H&nbsp;= Hit) will take different values depending on whether L is red, yellow or green (and likewise for P(H&nbsp;= Not&nbsp;Hit)). A person is, for example, far more likely to be hit by a car when trying to cross while the lights for perpendicular traffic are green than if they are red. In other words, for any given possible pair of values for H and L, one must consider the [[joint probability distribution]] of H and L to find the probability of that pair of events occurring together if the pedestrian ignores the state of the light.

However, in trying to calculate the '''marginal probability''' P(H&nbsp;= Hit), what is being sought is the probability that H&nbsp;= Hit in the situation in which the particular value of L is unknown and in which the pedestrian ignores the state of the light. In general, a pedestrian can be hit if the lights are red OR if the lights are yellow OR if the lights are green. So, the answer for the marginal probability can be found by summing P(H&nbsp;|&nbsp;L) for all possible values of L, with each value of L weighted by its probability of occurring.

Here is a table showing the conditional probabilities of being hit, depending on the state of the lights. (Note that the columns in this table must add up to 1 because the probability of being hit or not hit is 1 regardless of the state of the light.)

{| class="wikitable"
|+ Conditional distribution: <math>P(H\mid L)</math>
|-
! {{diagonal split header|H|L}}
! width="60"|Red
! width="60"|Yellow
! width="60"|Green
|-
! Not Hit
| align="center" | 0.99
| align="center" | 0.9
| align="center" | 0.2
|-
! Hit
| align="center" | 0.01
| align="center" | 0.1
| align="center" | 0.8
|}

To find the joint probability distribution, more data is required. For example, suppose P(L&nbsp;=&nbsp;red)&nbsp;= 0.2, P(L&nbsp;=&nbsp;yellow)&nbsp;= 0.1, and P(L&nbsp;=&nbsp;green)&nbsp;= 0.7. Multiplying each column in the conditional distribution by the probability of that column occurring results in the joint probability distribution of H and L, given in the central 2×3 block of entries. (Note that the cells in this 2×3 block add up to&nbsp;1).

{| class="wikitable"
|+ Joint distribution: {{tmath|P(H,L)}}
|-
! {{diagonal split header|H|L}}
! width="60"|Red
! width="60"|Yellow
! width="60"|Green
! width="60"|Marginal probability P(''H'')
|-
! Not Hit
| align="left" | 0.198
| align="left" | 0.09
| align="left" | 0.14
| align="left" | 0.428
|-
! Hit
| align="left" | 0.002
| align="left" | 0.01
| align="left" | 0.56
| align="left" | 0.572
|-
! Total
| align="left" | 0.2
| align="left" | 0.1
| align="left" | 0.7
| align="left" | 1
|}

The marginal probability P(H&nbsp;= Hit) is the sum 0.572 along the H&nbsp;= Hit row of this joint distribution table, as this is the probability of being hit when the lights are red OR yellow OR green. Similarly, the marginal probability that P(H&nbsp;= Not&nbsp;Hit) is the sum along the H&nbsp;= Not&nbsp;Hit row.