Create 15.tex

Author: Paciic

src/chapters/6/sections/moment_generating_functions/15.tex

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+(a). Since the MGF is uniquely determined from the distribution, and X and Y are identically distributed, \[M_{Y^2}(t) = M_{X^2}(t) = (1-2t)^{-\frac{1}{2}}.\]
+As X and Y are mutually independent, \[M_{W}(t) = M_{X^2+Y^2}(t) =  M_{X^2}(t) \cdot M_{Y^2}(t).\]
+Hence,
+\[M_{W}(t) = \frac{1}{(1-2t)}.\]
+
+(b). Exponential distribution, with \(\lambda =\frac{1}{2}\). Note that this is a special case (as mentioned in the question) of the Gamma distribution (Chapter 8).