# 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

## Related Questions

### Integral involving distance to the boundary

0  Asked on December 19, 2020 by isak

### In a compact metric space, a sequence with a certain property is convergent

1  Asked on December 19, 2020 by stakmod

### can you help with this exponential decay question?

2  Asked on December 19, 2020 by perfectoid

### Linear combinations of four-dimensional vectors

2  Asked on December 18, 2020 by sammy

### would relation induced subset break cartesian product – homomorphism of convex hull?

0  Asked on December 18, 2020 by peng-yu

### 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

### In how many ways can a group of six people be divided into: 2 equal groups? 2 unequal groups, if there must be at least one person in each group?

2  Asked on December 18, 2020 by user780357

### Area of $A:= lbrace (x,y) in mathbb{R}^2 : x^2+(e^x +y)^2leq1 rbrace$

1  Asked on December 18, 2020 by marvel

### 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

### logarithmic integral function and asymptotic expansion

1  Asked on December 17, 2020 by paris-lamp

### How to solve the following system : $begin{cases} 2x^2 +x + z = 0 \ (1-z) x^2 + 3x – y = 0 \ (2+y) x^2 + 4x – 2 = 0 end{cases}$?

2  Asked on December 17, 2020 by yoyo12

### Find smallest $sigma$-algebra with intersections

2  Asked on December 17, 2020 by puigi

### Does this PDE has solutions

1  Asked on December 17, 2020

### eigenfunction expansion to represent the best solution

0  Asked on December 17, 2020 by rubidium

### Smart method for computing the determinant of the matrix defined by $A_{ii}=a+b$ and $A_{ij}=a$?

4  Asked on December 17, 2020 by uri-toti

### Prove the de la Vallée-Poussin’s formula

1  Asked on December 16, 2020 by user834302

### Given 10 people: $P_1, P_2, ldots,P_{10}$ how many 6-member teams can be formed if

1  Asked on December 16, 2020 by scott