Recipe
Find a marginal distribution from a joint table
Use when: you have a joint probability table and need one variable's own distribution by itself
Goal: the complete list of probabilities for one variable, each obtained by summing across the joint table
Taught in Marginal distributions
Quick recipe
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1
Rows correspond to one variable's values, and columns to the other's. Sum WITHIN each row for the row variable's marginal, or WITHIN each column for the column variable's. Pick the direction that matches the table's own layout.
Example
In a weather/commute table, rain and clear label the rows, and late and on time label the columns. Weather's marginal comes from summing across each row; commute status's marginal comes from summing down each column.
Why: summing in the wrong direction adds together probabilities for different values of the variable you're not trying to isolate, producing a meaningless number.
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2
Add up every cell in that single row (or column) to get the marginal probability of that one value.
Example
P(\text{rain}) = 0.12+0.08=0.20
Why: every joint outcome consistent with "weather is rain," no matter what the commute status turns out to be, is exactly the cells in that row, so their total is the probability of rain alone.
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3
Do the same row-sum (or column-sum) for each of the variable's other values, one at a time, until every value has its own total.
Example
P(\text{clear}) = 0.06+0.74 = 0.80
Why: the marginal distribution isn't one number, it's the whole list of probabilities across every value the variable can take.
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4
Add up every marginal probability just found; the total should come out to exactly 1, the same validity condition every distribution must satisfy.
Example
P(\text{rain})+P(\text{clear}) = 0.20+0.80=1.00
⚡ Shortcut
When you need marginals for both values of a two-outcome variable, you don't have to sum twice. Once the first value's marginal is found, the second is just 1 minus it, since the two already add to 1. Here, once P(\text{rain})=0.20 is found, P(\text{clear})=1-0.20=0.80 follows immediately, without re-adding 0.06+0.74.
Why: this is the same check every valid distribution has to pass, and catching an arithmetic slip here, before the marginal gets used in a later computation such as conditioning, saves rework.
Common mistakes
- Reading a single joint cell as if it were the marginal, instead of summing the whole row or column it sits in.
- Summing in the wrong direction: across rows when the column variable was wanted, or down columns when the row variable was wanted.
- Stopping after finding the marginal for just one value, instead of every value of the variable.
- Skipping the sums-to-1 check on the finished marginal distribution, missing an arithmetic slip.