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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

1 Answers

One Answer

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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