# Probability between two events following exponential distribution

Two independent events A and B follow exponential distribution with parameter L: f(t)=Le^(-Lt) for t>=0. If X is the time where A occurs and Y the time where B occurs, calculate the probability P[X>=2*Y] meaning that A happens at least after double the time that B occurred.

How do I proceed? I need to integrate f(t) in the area where x>=2*y. Does that mean that 0<=y<=x/2?

• Random variables can follow exponential distribution but not events. – Kavi Rama Murthy Apr 10 at 9:56
• Correct. Thank you. – EmKal Apr 10 at 11:05

## 1 Answer

Assuming $$X$$ and $$Y$$ to be random variables, you need to integrate the joint distribution of $$(X,Y)$$ which is easy to compute as they are independent. Your integral limits are fine: $$0\leq x<\infty, 0\leq y\leq x/2$$. I think the final answer is $$1/3$$.

• Indeed, although using $0\leq y\leq \infty, 2y\leq x\leq \infty$ might be slightly easier to handle. – Graham Kemp Apr 10 at 10:21
• Worked like a charm, thank you. – EmKal Apr 10 at 12:19