Expectation problem: how many offers are expected until the owner sells his house?

Question

I am selling my house, and have decided to accept the first offer exceeding $£K$. Assuming that offers are independent random variables with common distribution function $F$, find the expected number of offers received before I sell the house.

MY ATTEMPT

This problem was proposed in the section of continuous random variables. However, I do not know how to solve it. It is worthy mentioning though that it is not a homework problem. Thanks in advance!

Answer 1

Let $p$ denote the probability that an offer exceeds $K$.

Let $\mu$ denote the expected number of offers received before you sell the house.

Then:$$\mu=p\cdot1+(1-p)\cdot(1+\mu)=1+(1-p)\mu\tag1$$leading to: $$\mu=\frac1p$$

Note that after an offer that does not make you sell the house you will return in the same situation again with exactly one more offer that was rejected. This declares the factor $1+\mu$ in $(1)$.

Here $p=P(X>K)=1-P(X\leq K)=1-F(K)$.

Another route for finding $\mu$ is application of geometric distribution.