What is the difference in
$\Pr(X>aY+b, Y>c)$ and $\Pr(X>aY+b|Y>c)$ in terms of the integration of the their pdf.
X and Y are indpendent RV.
2026-04-04 06:59:07.1775285947
Difference between integration of pdf in joint probability and conditional probability
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$$ P(X>aY + b, Y>c) = \int_{c}^\infty\int_{ay+b}^{\infty} dF_X(x) dF_Y(y)$$ where $dF_X(x)= f_X(x) dx$ and $dF_Y(y)= f_Y(y) dy$ if both variables are abs. continuous. $$ P(X>aY+b | Y>c) = \frac{P(X>aY+b , Y>c)}{P(Y>c)} $$ where $P(Y>c) = \int_c^\infty dF_Y(y)$. If X and Y are independent then $f_{X|Y}(x,y) = f_X(x)$.