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

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

Add your own answers!

Related Questions

Differentiation on tangent on a point

1  Asked on December 18, 2020 by kk6


P-NP for decision problems

1  Asked on December 18, 2020 by rasul_rza24


Max of $2$ independent random variables

2  Asked on December 17, 2020 by zestiria


time-varying dynamical system

2  Asked on December 17, 2020 by miss-q


Prove that the solutions of a linear system are the same when using RREF

0  Asked on December 17, 2020 by average_discrete_math_enjoyer


Find smallest $sigma$-algebra with intersections

2  Asked on December 17, 2020 by puigi


Prove the de la Vallée-Poussin’s formula

1  Asked on December 16, 2020 by user834302


Ask a Question

Get help from others!

© 2021 All rights reserved.