# Show that 𝑌1, 𝑌2, … are the arrival times of an inhomogeneous Poisson process. What is the intensity of this Poisson process?

Successive offers for my house are independent, identically distributed random variables $$X_1, X_2$$, ... having density function $$f$$ and distribution function $$F$$.

Let $$Y_1 = X_1$$, let $$Y_2$$ be the first offer exceeding $$Y_1$$, and, generally, let $$Y_{n+1}$$ be the first offer exceeding $$Y_n$$. Show that $$Y_1$$, $$Y_2$$, ... are the arrival times of an inhomogeneous Poisson process. What is the intensity of this Poisson process?

To show it is a poisson process, I am thinking we can show that $$Y_n-Y_{n-1}$$ which is the inter-arrival time is exponential distributed by showing memoryless. But how should I start... And I am confused by the relationship between $$X_n$$ and $$Y_n$$ right now. Please help. Thanks!

• I think you mean $Y_2$ and $Y_{n+1}$. – joriki Dec 6 '19 at 5:59
• Yes! thanks for pointing out – TMS Dec 6 '19 at 6:30
• See the solution on page 311-312 here: home.ustc.edu.cn/~zt001062/PTmaterials/… – Math1000 Jan 8 at 0:32