# Finding $P[X+Y > 1, X > 1]$

I’m trying to solve the following problem: i have two independent exponentially distributed r.v. $$X$$ and $$Y$$ both with $$lambda = 1$$. I want to know the probability $$P[X+Y > 1, X > 1]$$. Since they are independent, i wrote the joint p.d.f. as $$f_{X,Y}(x,y) = f_X(x) cdot f_Y(y) = e^{-x}cdot e^{-y}$$. Then i’ve tried to solve
$$int_{1}^{infty} int_{1 – x}^{infty} e^{-x}cdot e^{-y}, dy, dx$$
so,
$$int_{1}^{infty} e^{-x}cdot e^{x – 1} dx = e^{-1} cdot int_{1}^{infty} dx = infty$$
but i get as result $$infty$$. What am i doing wrong? I think the integral is correctly soved so the initial fomula of the joint may be wrong…

Mathematics Asked by damianodamiano on November 21, 2021

Since $$X$$ and $$Y$$ are positive random variables the event $$(X>1, X+Y>1)$$ is same as $$X>1$$. So the value is $$int_1^{infty} e^{-x} dx=frac 1 e$$.

The mistake in your calculation is the integral w.r.t $$y$$ should, start from $$0$$ and not $$1-x$$ because $$1-x <0$$.

Answered by Kavi Rama Murthy on November 21, 2021

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