A marginal probability gives the distribution of one variable after summing out the other variables in a joint distribution. For exhaustive, mutually exclusive values E₁,…,Eₙ, p(G) = Σᵢ p(Eᵢ, G). Adding every joint event in a row or column accounts for all situations in which the retained event can occur. section 5.4.1 section 6.1.1
Use marginal probabilities to:
- Read a variable’s overall distribution from a joint-probability table by summing across the other variable’s values. section 5.4.1
- Check normalization: marginal probabilities sum to 1 because they collectively cover every possible joint outcome. section 6.1.1
- Assess independence by comparing p(X, Y) with p(X)p(Y); equality holds for independent variables. section 6.1.1
- Supply the denominator in Bayes’ theorem, where the marginal probability is obtained by summing joint probabilities. section 6.1.3
Summing a joint table
Suppose distance G₁ has joint probabilities 0.05, 0.10, and 0.05 with weight categories E₁, E₂, and E₃. Then p(G₁) = 0.05 + 0.10 + 0.05 = 0.20. Summing the entries for E₂ across G₁, G₂, and G₃ similarly gives p(E₂) = 0.50. section 5.4.1
Table 5.6 Joint probability distribution with marginal probabilities shown
| Less than 60 kg (E1) |
Between 60 and 90 kg (E2) |
More than 90 kg (E3) |
Marginals for G’s |
|
|---|---|---|---|---|
| Less than 5 km (G1) |
p(E1, G1) = 0.25 × 0.2 = 0.05 |
p(E2, G1) = 0.5 × 0.2 = 0.1 |
p(E3, G1) = 0.25 × 0.2 = 0.05 |
p(G1) = 0.05 + 0.1 + 0.05 = 0.2 |
| Between 5 and 15 km (G2) |
p(E1, G2) = 0.25 × 0.6 = 0.15 |
p(E2, G2) = 0.5 × 0.6 = 0.3 |
p(E3, G2) = 0.25 × 0.6 = 0.15 |
p(G2) 0.15 + 0.3 + 0.15 = 0.6 |
| More than 15 km (G3) |
p(E1, G3) = 0.25 × 0.2 = 0.05 |
p(E2, G3) = 0.5 × 0.2 = 0.1 |
p(E3, G3) = 0.25 × 0.2 = 0.05 |
p(G3) = 0.05 + 0.1 + 0.05 = 0.2 |
| Marginals for Es |
p(E1) = 0.05 + 0.15 + 0.05 0.05 = 0.25 |
p(E2) = 0.1 + 0.3 + 0.1 = 0.5 |
p(E3) = 0.05 + 0.15 + = 0.25 |
Inspect the joint table with row and column marginals to trace each sum numerically. section 5.4.1
Table 6.1 Example population sizes and joint probability distribution for variables W = {E1, E2, E3} and H = {F1, F2, F3} weights and heights of adult Statsville residents), showing marginal probabilities
| Less than 60 kg (E1) |
Between 60 and 90 kg (E2) |
More than 90 kg (E3) |
Marginals for Fs |
|
|---|---|---|---|---|
| Less than 160 cm (F1) |
pop. = 20,000 p(E1, F1) = 0.2 |
pop. = 4,000 p(E2, F1) = 0.04 |
pop. = 2,000 p(E3, F1) = 0.02 |
pop. = 26,000; p(F1) = 0.2 + 0.04 + 0.02 = 0.26 |
| Between 160 cm and 183 cm (F2) |
pop. = 4,000 p(E1, F2) = 0.04 |
pop. = 40,000 p(E2, F2) = 0.4 |
pop. = 4,000 p(E3, F2) = 0.04 |
pop. = 48,000; p(F2) = 0.04 + 0.4 + 0.04 = 0.48 |
| More than 183 cm (F3) |
pop. = 2,000 p(E1, F3) = 0.02 |
pop. = 4,000 p(E2, F3) = 0.04 |
pop. = 20,000 p(E3, F3) = 0.2 |
pop. = 26,000; p(F3) = 0.02 + 0.04 + 0.2 = 0.26 |
| Marginals for Es |
p(E1) = 0.2 + 0.04 + 0.02 = 0.26 |
p(E2) = 0.04 + 0.4 + 0.04 = 0.48 |
p(E3) = 0.02 + 0.04 + 0.2 = 0.26 |
Total pop. = 100,000; Total prob = 1 |
Compare weight and height marginals with their joint cells to investigate dependence. section 6.1.1
