Question

Give an example of X being a uniformly distributed random vector on some set S⊂Rⁿ for some n≥2, for which the coordinate random variables X₁ ,…,Xₙ are not independent.

Answer 1

A uniformly distributed random vector is an example of a continuous random variable where the coordinate random variables may not be independent.

Step-by-step explanation:

A uniformly distributed random vector is an example of a continuous random variable that has equally likely outcomes over a specified set in n-dimensional Euclidean space. In this case, let's consider a set S that is a square in R².

Suppose we have X as a uniformly distributed random vector in S. Each coordinate random variable, X₁, X₂, ..., Xₙ, represents a coordinate of X. Now, for these coordinate random variables to not be independent, we can consider a case where S is a rectangle instead of a square. Specifically, let S be a rectangle that is longer along the X-coordinate axis. In this case, the X-coordinate random variables will be negatively correlated because if the X-coordinate deviates to the left, the probability of observing a larger Y-coordinate will be higher due to the elongated shape of the rectangle.