Let X be a random variable with PDF pX(x) =1/(squareroot 2 pie)1/x exp((log(x))^2/2)IR+(x), and let Y be a random variable with PDF pY (y) = pX(y)(1

Let X be a random variable with PDFpX(x) =1/(squareroot 2 pie)1/x exp(−(log(x))^2/2)IR+(x), and let Y be a random variable with PDFpY (y) = pX(y)(1 + wsin(2pie log(y)))IR+(y),where w is a constant and |w| <= 1.Notice the similarity of the PDF of Y to the PDF given in equation.a) Show that X and Y have different distributions.b) Show that for r = 1, 2, . . ., E(Xr ) = E(Y r).c) Notice that the PDF is that of the lognormal distribution, which, of course, is an absolutely continuous distribution. Now consider the discrete random variable Ya whose distribution, for givena > 0, is defined byPr(Ya = ae^k) = cae^((−k^2)/2)/a^−k, for k = 0, 1, 2, . . . ,where ca is an appropriate normalizing constant. Show that this discrete distribution has the same moments as the lognormal.Hint: First identify the support of this distribution. Then multiply the reciprocal of the rth moment of the lognormal by the rth of Ya.