Given that X N(0,1) then show that the pdf of Y=X2 is "\\chi" 12
Given that random variable "Y"is defined as "Y = X^2," where "X\\sim N(0, 1)."
Then for "y<0," "P(Y<y)=0" and
for "y\\geq 0,"
"P(Y<y)=P(X^2<y)=P(-\\sqrt{y}<X<\\sqrt{y})""=F_X(\\sqrt{y})-F_X(-\\sqrt{x})=F_X(\\sqrt{y})-(1-F_X(\\sqrt{y}))"
"=2F_X(\\sqrt{y})-1"
"f_Y(y)=\\dfrac{d}{dy}(2F_X(\\sqrt{y})-1)=2\\dfrac{dF_X(\\sqrt{y})}{dy}"
"=2\\dfrac{d}{dy}\\bigg(\\displaystyle\\int_{-\\infin}^{\\sqrt{y}}\\dfrac{1}{\\sqrt{2\\pi}}e^{-t^2\/2}dt\\bigg)"
"=\\dfrac{1}{\\sqrt{2}}(\\dfrac{1}{\\sqrt{\\pi}})e^{-y\/2}(\\dfrac{1}{\\sqrt{y}})"
Use "\\sqrt{\\pi}=\\Gamma(\\dfrac{1}{2})." Then
where "F" and "f" are the cdf and pdf of the corresponding random variables.
Then "Y=X^2\\sim\\chi_1^2" .
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