Question

5. Suppose X has the Rayleigh density otherwise 0, a. Find the probability density function for Y-X using Theorem 8.1.1. b. Use the result in part (a) to find E() and V(). c. Write an expression to calculate E(Y) from the Rayleigh density using LOTUS. Would this be easier or harder to use than the above approach? of variables in one dimension). Let X be s Y(X), where g is differentiable and strictly incr 1 len the PDF of Y is given by 1.1 (Change of variab PDFx, and let f trictly decresing).n g), in 1() The suppor of Y is all l)with z in the support of X. strictly increasing. The CDF of Y is Let g be he chain rule, the PDF of Y is o hy the fy (y) = fx (z) dr strictly decreasing is analogous. In that case the PDF ends up as proof to ich is nonnegative since 0 if g is strictly decreasing, Using 1 t of the theorem, covers both cases. tving the change of variables formula, we can choose whether to compute Mhen applying or to co we can do whichever is easier. Either way, in the end we should es the PDF of Y as a function of y. of variables formula (in the strictly increasing g case) is easy to remem- when written in the form br ich has an aesthetically pleasing symmetry to it. This formula also makes sense if think about units. For example, let X be a measurement in inches and Y = 2.54X br the conversion into centimeters (cm). Then the units of fx(x) are inches ad the units of fy (v) are cml, so it would be absurd to say something like Yv) = fr (x)" . But dz is measured in inches and dy is measured in cm, so fy(y)dy alfr()dr are unitless quantities, and it makes sense to equate them. Better yet, dr and fy (v)dy have probability interpretations (recall from Chapter 5 that izld is essentially the probability that X is in a tiny interval of length dz, cetered at z), which makes it easier to think intuitively about what the change of wiables formula is saying. e bext two examples derive the PDFs of two r.v.s that are defined as transforma- ot a standard Normal r v In the first example the change of variables formula xyies: in the second example it does not.

Answer 1

兀 1,14)

0. 3 E (y ar LOTUS