Understanding a Copula with an Interval-Restricted Marginal

Summary

The document asks how to express the joint probability that one variable is below a threshold while another lies within an interval. It compares the usual copula expression for two cumulative events with an attempted subtraction of joint probabilities, and questions a paper’s use of the interval probability as a copula marginal.

The response says the interval event has marginal probability given by the difference between the second variable’s cumulative probabilities at the interval endpoints, then says to insert the marginals into the copula. It also flags that the proposed subtraction does not represent the requested event. The answer is concise and offers no derivation involving transformed uniform variables or discussion of conditions such as continuity, so the underlying joint-probability step remains underexplained.

Key ideas

• A copula links marginal probabilities to a joint probability.
• The probability that a variable falls within an interval is expressed as a difference of cumulative probabilities.
• The response applies that interval probability as the second copula marginal.
• Subtracting two joint cumulative probabilities as proposed in the question does not directly express the requested event.
• The document does not provide a full derivation or discuss assumptions on the marginals.

Tags

# Where does this copula come from?


# Where does this copula come from?












In a paper I encountered the following notation

$$P(Z\leq z,u\leq Y\leq v)=C(F_{Z}(z),F_{Y}(v)-F_{Y}(u))$$

However I don't see why this holds in relation to uniform random variables. Usually $$P(Z\leq,Y\leq v)=P(F_{Z}(Z)\leq F_{Z}(z),F_{Y}(Y)\leq F_{Y}(v))=P(U_{1}\leq F_{Z}(z), U_{2}\leq F_{Y}(v))=C(F_{Z}(z),F_{Y}(v))$$

But probability above i would write $$P(Z\leq z,u\leq Y\leq v)=P(F_{Z}(Z)\leq F_{Z}(z),F_{Y}(Y)\leq F_{Y}(v))-P(F_{Z}(Z)\leq F_{Z}(z),F_{Y}(Y)\leq F_{Y}(u))$$ and then use copulas.

Can anyone explain to me where the copula $C(F_{Z}(z),F_{Y}(v)-F_{Y}(u))$ comes from in terms of uniform random variables?

## Answer by Aksakal almost surely binary (score 2)

https://quant.stackexchange.com/a/10744

if you agree that the marginal probability $P(u\le Y\le v)=F_Y(v)-F_Y(u)$, then your formula follows immediately, because next you simply plug the marginals into the copula.

your 3rd equation for the joint probabilities is incorrect for $P(Z\le z,u\le Y\le v)$, I'm not sure where you got it from

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