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Question 6.2: Find the PDF of the random variable Y=aX², where X is the st......

Find the PDF of the random variable Y=aX^2, where X is the standard normal random variable and \,\displaystyle a>0.

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The PDF of X is given by

\,\displaystyle f_X\left (x\right )=\frac{1}{\sqrt{2 \pi}} e^{-x^2 / 2} \quad-\infty<x<\infty

which is an even function. Thus, from the preceding result we have that

\,\displaystyle f_Y\left (y\right )=\frac{f_X\left (\sqrt{y / a}\right )}{\sqrt{a y}}=\frac{e^{-y / 2 a}}{\sqrt{2 \pi a y}} \quad y>0

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