We have two rvs $X$ and $Y$ and calculated $$Pr\{X > x| Y = y\} = g(x) \tag{1}\label{eq1}$$ Can we argue that based on \eqref{eq1} $Pr\{X > x\}=g(x)$ and now since $Pr\{X > x| Y = y\} = Pr\{X > x\}$ rvs $X$ and $Y$ are statistically independent?
2026-05-15 05:47:16.1778824036
Find distribution and statistical independence based on calculations of conditional probability
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Yes, if we have $\forall x~\forall y~.\mathsf P(X\,{>}\,x\mid Y\,{=}\,y)=g(x)$, then that is a valid conclusion.
Being a monovariate function of $x$ for all values of $x$ and $y$ clearly indicates that the conditional probability is independent of the realisation of $Y$, and therefore that $X$ and $Y$ are independent.