Problem 2.7 (Linear Transformations of the Data). Consider a data set with n values X = (x1,...,xn), with mean r, variance , and standard deviation r. If build a new data set of n elements = (y1,..., Yn) where each element is obtained by applying a function y = f(x) (a "transformation", if you prefer) to the elements of X, this operation will change the mean, variance and standard deviation of the set. When the function is linear, such statistics for y can be computed in a (relatively) simple manner. a. (Shift by a constant c.) Suppose that y = x; +c, for all i = 1,...,n, where c is a constant. (i) Express y in terms of and c. (ii) Also, express in terms of and c. Actually, do you need c for the latter? (What happens to the histogram of X if you add the same constant c to each value xi?) b. (Scaling by a constant a.) Suppose now that yi = axi, for all i = 1,...,n, where a is a constant. (i) Express y in terms of x and a. (ii) Also, express in terms of and a. (iii) Finally, express y in terms of and a (this is not as trivial as you may think: what happens for a < 0?). c. (General linear transformations.) Finally, let's put everything together and suppose that y; = axi+c, for all i = 1,..., n, where a and c are constants. (i) Express y in terms of and the constants a and c. (ii) Also, express in terms of 2, a and c. (iii) Finally, express y in terms of x, a and c.