http://www.ask.com/wiki/Marginal_probability
Real-world example
Imagine for example you want to compute the probability that a pedestrian will be hit by a car while crossing the road at a pedestrian crossing. Let H be a discrete random variable describing the probability of being hit by a car while walking over the crossing, taking one value from { hit, not hit }. Let L be a discrete random variable describing the probability of the traffic light state at a given moment, taking one from { red, yellow, green }.
Realistically, H will be dependent on L. That is, P(H = hit) and P(H = not hit) will take different values depending on whether L is red, yellow or green. You are, for example, far more likely to be hit by a car if you try to cross while the lights are green than if they are red. In other words, for any given possible pair of values for H and L, you must feed them into the joint probability distribution of H and L to find the probability of that pair of events occurring together.
However, in trying to calculate the marginal probability P(H=hit), what we are asking for is the probability that H=hit, where we don't actually know the particular value of L. In general you can be hit if the lights are red OR if the lights are yellow OR if the lights are green. So in this case the answer for the marginal probability can be found by summing P(H,L) = P(hit,L) for all possible values of L.
Here is a table showing the conditional probabilities of being hit, depending on the state of the lights. (Note that due to the dependence, only the columns in this table must add up to 1).
| Conditional distribution: P(H|L) | |||
|---|---|---|---|
| L=Red | L=Yellow | L=Green | |
| H=Not Hit | 0.99 | 0.9 | 0.2 |
| H=Hit | 0.01 | 0.1 | 0.8 |
To find the joint probability distribution, we need more data. Let's say that P(L=red) = 0.7, P(L=yellow) = 0.1, P(L=green) = 0.2. Multiplying the columns in the conditional distribution the by the appropriate values, we find the joint probability distribution of H and L. (Note that the cells in this table do now add up to 1).
| Joint distribution: P(H,L) | ||||
|---|---|---|---|---|
| L=Red | L=Yellow | L=Green | Marginal probability | |
| H=Not Hit | 0.693 | 0.09 | 0.04 | 0.823 |
| H=Hit | 0.007 | 0.01 | 0.16 | 0.177 |
| Total: | 1 | |||
The marginal probability P(H=Hit) is the sum of the bottom row, as this is the probability of being hit when the lights are red OR yellow OR green. Similarly, the marginal probability that P(H=Not Hit) is the sum of the top row. It is important to interpret this result correctly. The chance of being hit by a car when you cross the road is obviously a lot less than 17.7%. However, what this figure is actually saying is that if you were to ignore the state of the traffic lights and cross the road no matter their color, you would have a 17.7% risk of being hit by a car. This seems more reasonable.
General cases
For multivariate distributions, formulae similar to those above apply with the symbols X and/or Y being interpreted as vectors. In particular, each summation or integration would be over all variables except those contained in X.
See also
References
- ^ Trumpler and Weaver (1962), pp. 32–33.
Bibliography
- Everitt, B. S. (2002). The Cambridge Dictionary of Statistics. Cambridge University Press. ISBN 0-521-81099-x.
- Trumpler, Robert J. and Harold F. Weaver (1962). Statistical Astronomy. Dover Publications.
