# Expected value of $E[X_n]$

I have the following problem:

Suppose that a point $$X_1$$ is chosen from a uniform distribution on the interval $$[0, 1]$$, and that after the value $$X_1=x_1$$ is observed, a point $$X_2$$ is chosen from a uniform distribution on $$[x_1, 1]$$. The variables $$X_3, X_4,\ldots$$ are generated in the same way. In general, for $$j=1, 2,\ldots$$, after $$X_j=x_j$$ is observed, $$X_{j+1}$$ s chosen from a uniform distribution on $$[x_j, 1]$$. Calculate $$E[X_n]$$.

So, I get the the probability for $$X_n$$ to be equal to some value $$a\in [x_{n-1}, 1]$$ is $$\dfrac{1}{1-x_n}$$, given that $$X_1=x_1, X_2=x_2,\ldots X_{n-1}=x_{n-1}$$. But I don't really know how to continue to get the expected value from here. Any help would be really appreciated.

• $\dfrac{1}{1-x_n}$ is a density rather than a probability Jan 9 '20 at 8:57

## 1 Answer

Hint : \begin{align*} \mathbb E[X_n] &= \mathbb E[\mathbb E[X_n|X_{n-1}]]\\ &= \mathbb E\left[ \frac{1+X_{n-1}}{2} \right]\\ &= \frac{1}{2} + \frac{\mathbb E[X_{n-1}]}{2} \end{align*}