Final answer:
The condition E(Y|X) = E(Y) implies that X and Y are uncorrelated with a covariance of zero. To construct their joint distribution, we could independently assume distributions for both, such as uniform, and their joint probability is the product of their individual probabilities. The correlation coefficient would also be zero, indicating no linear relationship.
Step-by-step explanation:
If we are given that E(Y|X) = E(Y), we can deduce that X and Y are uncorrelated because the expected value of Y does not depend on the value of X. The covariance between two variables, Cov(X, Y), is a measure of the joint variability of those variables. Given the provided condition, we know that Cov(X, Y) is zero. Furthermore, the coefficient of correlation, which measures the strength of association, will also be zero (Corr(X, Y) = 0) if there is no linear relationship between X and Y.
To construct the joint distribution of X and Y, we can assume any distribution for X and Y independently because their independence would imply that the joint probability P(X and Y) can be found by multiplying their individual probabilities, following P(X and Y) = P(X)P(Y). One common assumption for simplicity could be to assume that both X and Y have uniform distributions. From the information provided, we can define the theoretical distribution of X as X ~ U(0,1), where U denotes the uniform distribution. The parameters for a uniform distribution are µ (mean) and σ (standard deviation).
To summarize, given that the variables are independent, for any function of X, it will not affect the distribution of Y, thus implying that every function of X can be viewed as independent of Y.