Consider an iid sample X_1, X_2, ldots , X_ n stackrel{iid}{sim } mathbf{P} that has been reordered as X_{(1)} leq X_{(2)} leq ldots leq X_{(n)} where n is very large. In the problems below, we have chosen a different distribution for mathbf{P} and compared the empirical quantiles to the standard Gaussian quantiles using a QQ plot. Recall that
the Laplace distribution ext {Lap}(lambda ) with parameter lambda > 0 is the continuous probability distribution with density f_lambda = frac{lambda }{2} e^{-lambda |x|}, and
the Cauchy distribution is the continuous probability distribution with density g(x) = frac{1}{pi } frac{1}{1 + x^2}.
(These were also introduced in Lecture 12. )
For each plot below, match the QQ plot with the correct distribution for mathbf{P}. Hint: Each possible distribution will be an answer choice exactly once, so you should use the process of elimination.
Hint: You may use computational tools to graph the pdf of the possible distributions of mathbf{P}.