Question

2X x 20 5 pt. a. Find the cdf and pdf of Y in terms of the cdf and pdf of X. of Y when X is a Gaussian random variable with zero mean and variance-4

Answer 1

$a.~CDF~of~Y~is\\ F_Y(y)=P(Y\leq y)=P(3X\leq y)=F_X(y/3)~if~y<0\\ =P(3X<0)&plus;P(0\leq 2X\leq y)~if~y\geq 0\\ =F_X(0)&plus;F_X(y/2)-F_X(0)=F_X(y/2)~if~y\geq 0\\ PDF~of~Y~is\\ f_Y(y)=\frac{\mathrm{d} F_Y(y)}{\mathrm{d} y}\\ =\frac{1}{3}f_X(y/3)~if~y<0\\ =\frac{1}{2}f_X(y/2)~if~y\geq 0\\ b.~The~pdf~of~X~is\\ f_X(x)=\frac{1}{2\sqrt{2\pi}}\exp\left(-\frac{x^2}{8} \right ),~-\infty<x<\infty\\ The~pdf~of~Y~is\\ f_Y(y)=\frac{1}{6\sqrt{2\pi}}\exp\left(-\frac{y^2}{72} \right )~if~y<0\\ =\frac{1}{4\sqrt{2\pi}}\exp\left(-\frac{y^2}{32} \right )~if~y\geq 0\\$