Final answer:
Direct determination of the joint cumulative distribution function (CDF) for the random vector (x, y) is not possible from the provided options without additional information on the joint probability density function and the relationship between X and Y.
Step-by-step explanation:
The joint cumulative distribution function (CDF) of the random vector (x, y) cannot be directly determined from the provided integral ∫[0 to a] ∫[0 to b] e^(-y²) dy dx. However, we can determine if option a or b is the correct expression for the joint CDF by examining the properties of joint CDFs and evaluating the given expressions.
Option a describes the result of an integral of e^(-y²) over the range y=0 to b and x=0 to a. This double integral does not represent a standard form for the joint CDF as it lacks integration of a proper joint probability density function (pdf) which typically depends on both variables x and y.
Option b states an expression 1 - e^(-b²) - e^(-a²) + e^(-(a² + b²)). This expression suggests subtraction of independent marginal CDFs and addition of a joint term, but without further context on the nature of the random variables X and Y and their interdependence, we cannot confirm this as the joint CDF.
Without additional information, we cannot definitively determine the correct joint CDF for the random vector (x, y) from the given options. Additional details about the joint pdf and the relationship between X and Y are needed to find the correct joint CDF.