Let $T$ the survival time of patients with $F(.)$ and $S_T(x)=P(T>x)$ the cumulative distribution function and the survival function. Take a covariate $Z$ such that $$Z=\begin{cases}0 \qquad group 0\\1 \qquad group1 \end{cases}$$ Then
$S_0(x)=P(T>x;Z=0)$
$S_1(x)=P(T>x;Z=1)$
Taking $S_0(\phi x)=S_1(x)$ for $\phi\in \mathbb{R}$, find the relationship between the probability density functions and the risk functions.
I don't know what I need to do in this question. Let $$S_1(x)=P(T>x)=1-P(T\leq x)$$ then $$\frac{\partial}{\partial x}S_1(x)=-f_T(x)$$ So if $S_0(\phi x)=S_1(x)$
$$S_1(x)=S_0(\phi x)\Rightarrow P(T>\phi x)=P(T>x)$$
but it's true just if $\phi=1$. Anyone have any idea what I need to do? It can be a typo in $T$ where it should be $T_1$ and $T_2$?