Which iid variables give a normal variable, when multiplied? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T03:38:45Z http://mathoverflow.net/feeds/question/6948 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/6948/which-iid-variables-give-a-normal-variable-when-multiplied Which iid variables give a normal variable, when multiplied? Arkadiusz Paterek 2009-11-27T11:13:45Z 2009-11-27T15:46:18Z <p>Hello, I hope you'll find my riddle interesting.</p> <p>Z = XY<br /> Z ~ N(0, 1)<br /> X, Y are iid random variables (independent, identically distributed). We assume X and Y are symmetric.<br /> What is the distribution of X and Y?</p> <p>It will be enough to find the answer for Z(+) = X(+) Y(+), where Z(+)'s distribution is the positive half of N(0,1), and X(+) and Y(+) are positive.</p> <p>I am interested in knowing the pdf of X,Y, if it has a closed form.</p> http://mathoverflow.net/questions/6948/which-iid-variables-give-a-normal-variable-when-multiplied/6959#6959 Answer by Jonathan Kariv for Which iid variables give a normal variable, when multiplied? Jonathan Kariv 2009-11-27T15:46:18Z 2009-11-27T15:46:18Z <p>First thought is to take U=ln[Z(+)]= ln[X(+)]+ln[Y(+)]. Work out the characteristic function of U and then square root it to get the characteristic function of ln[X(+)]. Sure the intergrals will be horribly messy though.</p>