Some elementary proofs of the normality of XY(X2 + Y2)12 when X and Y are normal

✍ Scribed by L. Baringhaus; N. Henze; D. Morgenstern

• Book ID: 103930392
• Publisher: Elsevier Science
• Year: 1988
• Tongue: English
• Weight: 119 KB
• Volume: 15
• Category: Article
• ISSN: 0898-1221
• DOI: 10.1016/0898-1221(88)90038-7

✦ Synopsis

We give three elementary and short proofs of the following statement: if X and Y are independent, normally distributed random variables with zero means and variances o 2 > 0 and 32 > 0, respectively, then the random variable XY/(X2+ y2)1/2 is also normal with mean zero and standard deviation ~/(¢ + z). No knowledge of the theory of stable laws is needed.