Question

8. (11 pts) Use the Transformation Method on this problem (be sure to verify that the function hy) is increasing or decreasing over the domain of y, either by graphing h(y) or by using differential calculus): A fluctuating electric current X may be considered a uniformly distributed random variable over the interval (6, 10). Find the probability density function of the power P which can be expressed as P = 2X2

Answer 1

Given a fluctuating electric current $X$ has uniform distribution $X\sim U\left ( 6,10 \right )$ . So $f_X\left ( x \right )=\frac{1}{10-6};6<X<10$ So We have to find the PDF of the transformation,

$Y=2X^2\Rightarrow X= \sqrt{\frac{Y}{2}}$. We know the PDF of the transformation. $Y=g\left ( X \right )$ is

$f_Y\left ( y \right )=f_X\left ( g^{-1}\left (y \right ) \right )\left | \frac{\mathrm{d} \left ( g^{-1}y \right )}{\mathrm{d} y} \right |\\$. If inverse function $X=g^{-1}\left ( Y \right )$ exists.

Plugging in functions,

$f_Y\left ( y \right )=f_X\left ( \sqrt{\frac{y}{2}} \right )\left | \frac{\mathrm{d} \sqrt{\frac{y}{2}}}{\mathrm{d} y} \right |\\ f_Y\left ( y \right )=\frac{1}{16}\sqrt{\frac{2}{y}}\\$

The PDF of power $Y=P=2X^2$. is

${\color{Blue} f_Y\left ( y \right )=\frac{1}{16}\sqrt{\frac{2}{y}};72<Y<200}$