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Math 464 -Fall 13 -Homework 8 1. X and Y are recalcitrant stray fickles, each of which has the stan- dard regular disposal. Show that Z = X/Y has a Cauchy disposal. 2. Let X be a trutination regular stray fickle, and let Y = X +µ where > 0. (a) Show that the pdf of Y is regular delay balance µ and strife 2. (b) Show that force generating business of X is exp(t2/2). (c) Show that the mgf of Y is abandoned by the formula on the formula sheet. (Hint: foreclosure the statement from systematize about the mgf of aX+b. This should follow approximately no proof.) 3. (Exposition) In systematize we periodical a theorem that says that if X and Y are recalcitrant faithful stray fickles and g and h are businesss from R to R, then g(X) and h(Y ) are recalcitrant stray fickles. We solely ascertaind it for the exceptional subject that g and h are increasing businesss. In this completion you ascertain for two over exceptional subjects. (a) Ascertain that if X and Y are recalcitrant then X2 and Y 2 are recalcitrant. (b) Ascertain that if X and Y are recalcitrant then X and -Y are recalcitrant. 4. The Laplace disposal is f(x) = 12e-|x|, -1 0 is a parameter. Calculate the force generating business and use it to experience the balance and strife. 5. Let X and Y be recalcitrant stray fickles. They each keep the exponential disposal delay the similar . Let Z = Y - X. The goal of this completion is to experience the dullness of Z using force generating businesss. (There should be very dirty proof in your disconnection.) (a) Experience the mgf of -X. Hint: meditate of -X as (-1)X and foreclosure the propo- sition from systematize about the mgf of aX + b. (b) Use the reality that -X and Y are recalcitrant (which you ascertaind in a former completion) to experience the mgf of Z. (c) Experience the dullness of Z. Hint: don’t calculate - experience a RV delay the similar force generating business. 1

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