I am really struggling with getting an intuition about copulas. I have red many articles and I am stuck at what is the concept/idea behind it.
For example if I have two random variables X and Y and I want to use copula to come up with their joint CDF. Suppose that both X and Y are normally distributed and I want to know what their joint CDF is. How does copula help me do that? Where does inverse transform theory come in?
Also what if they have different distributions? (let's say one is normally distributed and the other one is student's t distributed).
2026-02-23 15:48:43.1771861723
How can I get an intuition about Copula?
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Copula allows you to work with individual marginal CDF for each random variable (RV), instead of dealing with the joint distribution of all RVs at one time.
This is achieved by creating a joint CDF on a series of Uniform [0,1] RVs, where each Uniform RV is essentially the CDF of the underlying original RV. Note a function (including CDF) of an RV is still an RV. By doing this, an otherwise complex joint CDF of whatever distributions is transferred to a much simpler presentation of uniform RV and original marginal RV.
I feel the key to understand this is the probability integral transform, which shows why a CDF follows Uniform [0,1]. See https://en.wikipedia.org/wiki/Probability_integral_transform. I found this a little hard to appreciate, so I drew a chart below. I am not 100% sure so any comments are welcome.