Marginal Probability Distribution
marginal probability distribution
- is a probability distribution of n-1 variables formed by calculating the subset of a multivariate probability distribution of n variables, the resulting probability distribution of n-1 variables can be either:
- univariate probability distribution (1-dimensional probability distribution)
- multivariate probability distribution (multi-dimensional probability distribution)
Discrete Marginal Probability Distribution
- 𝐏(𝑋) = ∑𝑦∊𝑌 [ 𝐏(𝑋, 𝑌=𝑦) ]
- 𝐏(𝑋, 𝑌) = ∑𝑧∊𝑍 [ 𝐏(𝑋, 𝑌, 𝑍=𝑧) ]
- 𝐏(𝑋) = ∑𝑧∊𝑍 ∑𝑦∊𝑌 [ 𝐏(𝑋, 𝑌=𝑦, 𝑍=𝑧) ]
Continuous Marginal Probability Distribution
- 𝐏(𝑋) = -∞∫∞ 𝐏(𝑋, 𝑌) 𝑑𝑦
- 𝐏(𝑋, 𝑌) = -∞∫∞ 𝐏(𝑋, 𝑌, 𝑍) 𝑑𝑧
- 𝐏(𝑋) = -∞∫∞ [ -∞∫∞ 𝐏(𝑋, 𝑌, 𝑍) 𝑑𝑧 ] 𝑑𝑦
Discrete Example Joint PMF to Marginal PMF
| 𝑌 | 𝐏(𝑋) | ||||
| 𝑦=0 | 𝑦=1 | 𝑦=2 | 𝑦=3 | |||
𝑋 | 𝑥=0 | 0.20 | 0.20 | 0.05 | 0.05 | 0.50 |
| 𝑥=1 | 0.20 | 0.10 | 0.10 | 0.10 | 0.50 | |
| 𝐏(𝑌) | 0.40 | 0.30 | 0.15 | 0.15 | 1.00 | |
- adding row-wise joint probabilities we get the marginal probability mass function 𝐏(𝑋)
- adding column-wise joint probabilities we get the marginal probability mass function 𝐏(𝑌)
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