In the 1880s, Francis Galton studied the inheritance of physical characteristics. Galton found that the sons of tall men tended to be taller than average, but shorter than their fathers. Similarly, sons of short men tended to be shorter than average, but taller than their fathers. Thus, the average heights of the sons were closer to the mean height of the population, regardless of whether the fathers were taller or shorter than average. From these observations, one might conclude that the variability of height decreases over successive generations, both tall persons and short persons tend to be eliminated, and the population “regresses” toward some average height. This conclusion is an example of the regression fallacy. In this problem you will prove that the regression fallacy arises in the bivariate normal distribution even when both coordinates have the same variance. In particular, assume that the vector (X1, X2) has the bivariate normal distribution with common mean μ, common variance σ2, and positive correlation ρ < 1. Prove that E(X2|X1) is closer to μ than X1 is to μ for every value X1. (This occurs despite the fact that X1 and X2 have the same mean and the same variance.)