Consider the following contrived data set for the variables X1, X2, and X3, where the question marks indicate missing data:

(a) Using available cases (and recomputing the means and standard deviations for each pair of variables), find the pairwise correlations among the three variables and explain why the correlations are not consistent with each other.

(b) Compute the correlation between X1 and X2 using means and standard deviations computed separately from the valid observations for each variable. What do you find?

(c) 'Show that the available-case correlation matrix among the variables X1, X2, and X3 is not positive semidefinite.