probability density function of $c\times y$

Question

I have been given a pdf by $f(y)=2y$ for $0<y<1$. Now I am asked to find the pdf for $c\times y$ where $c$ is some constant $c\in \mathbb{R}_{+}$.

My initial thought process was that $f(c\times y)=c\times f(y)$. But this is wrong, because when I evaluate the integral

$\int_{0}^{1}(c\times f(y))dy=c$ instead of $1$.

So can anyone help me with this? Thanks in Advance.

Answer 1

Start by finding the CDF of $Z=cY$. Then $Z\in(0,c)$.

$P(Z\le z)=P(cY\le z)=P(Y\le z/c)=F_Y(z/c)$

Now differentiate it with respect to $z$ to obtain PDF,$$f_Z(z)=\frac{d}{dz}[F_Y(z/c)]=\frac1cf_Y(z/c)$$