In the study of multivariable probability distributions, especially in engineering applications involving random variables, we often deal with joint probability distributions. However, in many cases, we are interested in understanding the behavior of individual variables without considering the full joint behavior. This leads us to the concept of marginal distributions. Marginal distributions are essential tools in fields such as signal processing, reliability engineering, communication systems, and more. They allow for the analysis of individual variables by "marginalizing" or integrating out other variables.

Before diving into marginal distributions, let us briefly recall the idea of joint distributions. If 𝑋 and 𝑌 are two continuous random variables, their joint probability density function (pdf) is denoted by 𝑓 (𝑥,𝑦), which satisfies:
• 𝑓 (𝑥,𝑦) ≥ 0
• ∬ 𝑓 (𝑥,𝑦) 𝑑𝑥 𝑑𝑦 = 1
This function gives the probability density of the pair (𝑋,𝑌) taking on particular values.

A marginal distribution is the probability distribution of one variable irrespective of the others. For two random variables 𝑋 and 𝑌 with joint pdf 𝑓 (𝑥,𝑦), the marginal pdfs are defined as:
• Marginal pdf of 𝑋:
∞
𝑓 (𝑥) = ∫ 𝑓 (𝑥,𝑦) 𝑑𝑦
−∞
• Marginal pdf of 𝑌:
∞
𝑓 (𝑦) = ∫ 𝑓 (𝑥,𝑦) 𝑑𝑥
−∞
This process is called marginalization because we are "eliminating" the effect of the other variable by integration.

If 𝑋 and 𝑌 are discrete random variables with a joint probability mass function (pmf) 𝑝 (𝑥,𝑦), then:
• Marginal pmf of 𝑋:
𝑝 (𝑥)= ∑ 𝑝 (𝑥,𝑦)
• Marginal pmf of 𝑌:
𝑝 (𝑦) = ∑ 𝑝 (𝑥,𝑦)

Marginal distributions tell us the probability behavior of a single variable while ignoring the other variables. For example, if 𝑋 represents the temperature and 𝑌 represents the pressure in a system, the marginal distribution 𝑓 (𝑥) tells us how temperature behaves overall, regardless of the pressure.

• Signal Processing: Analysis of individual signals in joint time-frequency representations.
• Reliability Engineering: Estimating failure rates when multiple causes are modeled.
• Communication Systems: Isolating signal behaviors over noisy channels.
• Machine Learning: Feature distribution analysis for probabilistic models.

Problem: Given the joint pdf:
6𝑥𝑦, 0 < 𝑥 < 1,0 < 𝑦 < 1
𝑓 (𝑥,𝑦) = {
0, otherwise
Find the marginal distributions 𝑓 (𝑥) and 𝑓 (𝑦).
Solution:
1. Marginal pdf of 𝑋:
1 1 𝑦2 1 1
𝑓 (𝑥) = ∫ 6𝑥𝑦 𝑑𝑦 = 6𝑥∫ 𝑦 𝑑𝑦 = 6𝑥[ ] = 6𝑥⋅ = 3𝑥
2 2
0 0 0
Valid for 0 < 𝑥 < 1
2. Marginal pdf of 𝑌:
1 1 𝑥2 1 1
𝑓 (𝑦) = ∫ 6𝑥𝑦 𝑑𝑥 = 6𝑦∫ 𝑥 𝑑𝑥 = 6𝑦[ ] = 6𝑦 ⋅ = 3𝑦
2 2
0 0 0
Valid for 0 < 𝑦 < 1

• The marginal distributions are themselves valid probability distributions:
∫𝑓 (𝑥) 𝑑𝑥 = 1, ∫𝑓 (𝑦) 𝑑𝑦 = 1
• From marginal pdfs, one cannot reconstruct the joint pdf unless variables are independent.

If 𝑋 and 𝑌 are independent, then:
𝑓 (𝑥,𝑦) = 𝑓 (𝑥)⋅𝑓 (𝑦)
This condition can be tested by comparing the joint pdf with the product of marginal pdfs.

For three variables 𝑋,𝑌,𝑍, the marginal pdf of 𝑋 is:
∞ ∞
𝑓 (𝑥) = ∫ ∫ 𝑓 (𝑥,𝑦,𝑧) 𝑑𝑦 𝑑𝑧
−∞ −∞

• Marginal distributions provide insights into individual variables in multivariate distributions.
• They are derived by integrating (or summing) the joint distribution over the other variables.
• In engineering, marginal distributions are used in probabilistic modeling and signal analysis.
• Understanding marginals is key to simplifying complex systems and focusing on specific variables of interest.