How to Solve Continuous Random Variables — Univariate RV — distribution of a transform Y = X^2 Questions on Exam P
Sample Practice Problem
ID: #QCO4J
A laboratory's output \(X\) is continuous with density \(f_{X}\). A calibrated output is \(Y=2X\). Which expression is the density of \(Y\)?
📖 Worked Solution & Strategy
The calibration is a one-to-one increasing transformation because its scale factor is positive. Its CDF follows by transforming an inequality, and differentiating that CDF supplies the required Jacobian factor for the density. Because \(Y=2X\) with \(2>0\), the cumulative distribution function (CDF) of \(Y\) is \(F_{Y}(y)=\Pr(Y\le y)=\Pr(2X\le y)=F_{X}\!\left(\dfrac{y}{2}\right)\). Differentiating with respect to \(y\) yields the probability density function \(f_{Y}(y)=\dfrac{d}{dy}F_{X}\!\left(\dfrac{y}{2}\right)=\dfrac{1}{2}f_{X}\!\left(\dfrac{y}{2}\right)\). Thus the density identity is \(f_{Y}(y)=\dfrac{1}{2}f_{X}\!\left(\dfrac{y}{2}\right)\).