Marginal Distributions and Joint Probability in Simple Terms


When exploring the probability of events, such as bond defaults, it's critical to grasp that knowing the individual likelihoods, or marginal distributions, of each bond defaulting does not inherently inform us about the likelihood of multiple bonds defaulting at the same time. This concept is key because even if two sets of bonds have identical marginal probabilities, their joint probabilities can differ dramatically based on how their defaults are correlated.


Marginal distributions tell us about the individual behavior of each variable (in this case, each bond's default probability) without regard to any other variables. For example, if Bond 1 and Bond 2 both have a 50% chance of defaulting, these are their marginal probabilities. However, these probabilities don't reveal how Bond 1's default might be influenced by or correlate with Bond 2's default.


The joint distribution, however, tells us about the probability of various combinations of these events occurring simultaneously. If bonds are independent, as in Scenario 1, their joint probability distribution simply reflects the product of their marginal probabilities. Here, the joint probability table reflects this independence:


B1 B2 0 (No Default) 1 (Default) P(B1)
0 0.25 0.25 0.5
1 0.25 0.25 0.5
P(B2) 0.5 0.5

However, if we consider Scenario 2, where Bond 3 and Bond 4 are perfectly correlated, the joint probability distribution changes dramatically, despite the marginal probabilities remaining the same as in the first scenario.


In this case, the default of Bond 3 guarantees the default of Bond 4, and vice versa, leading to a joint distribution that focuses on the extremes, with no probability spread between them:


B3 B4 0 (No Default) 1 (Default) P(B3)
0 0.5 0 0.5
1 0 0.5 0.5
P(B4) 0.5 0.5

This contrast between Scenarios 1 and 2 illustrates a fundamental statistical truth: identical marginal distributions do not guarantee the same joint distribution. The underlying dependency structure, which can be modeled using copulas, plays a crucial role in shaping the joint distribution.



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About the Author

 

 Florian Campuzan is a graduate of Sciences Po Paris (Economic and Financial section) with a degree in Economics (Money and Finance). A CFA charterholder, he began his career in private equity and venture capital as an investment manager at Natixis before transitioning to market finance as a proprietary trader.

 

In the early 2010s, Florian founded Finance Tutoring, a specialized firm offering training and consulting in market and corporate finance. With over 12 years of experience, he has led finance training programs, advised financial institutions and industrial groups on risk management, and prepared candidates for the CFA exams.

 

Passionate about quantitative finance and the application of mathematics, Florian is dedicated to making complex concepts intuitive and accessible. He believes that mastering any topic begins with understanding its core intuition, enabling professionals and students alike to build a strong foundation for success.