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
|
𝐏(𝑋, 𝑌) |
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)