Conditional expectation of a sequence of random variables

Question

$X_1 \sim \mathrm{Unif}(0,1)$

if $X_1=x_1$, $X_2 \sim \mathrm{Unif}(x_1,x_1+1)$

if $X_2=x_2$, $X_3 \sim \mathrm{Unif}(x_2,x_2+1)$

for $n \geq4$, $X_n$ is defined the same way. How do I calculate $E(X_n)$?

I know that the $E(X)=\frac{b+a}2$ for $X\sim \mathrm{Unif}(a,b)$ But this doesn't seem as simple as using that because $E(X_n)$ depends on the value of $X_{n-1}$ and so on. Any help would be greatly appreciated.

Answer 1

$X_{n+1}$ is uniform on $(X_n,X_n +1)$ so $E(X_{n+1} |X_n)=X_n +\frac 1 2$. Taking expectation on both sides we get $EX_{n+1}=EX_n+\frac 1 2$. Hence, by induction, $EX_n =\frac n 2$.

Answer 2

Guide:

$$E[X_1]=\frac12$$

$$E[X_2]=E[E[X_2|X_1]]=E[X_1+\frac12]=1$$

Go on with this procedure for a few more terms and think of a conjecture.

$$E[X_n]=E[E[X_n|X_{n-1}]]=E[X_{n-1}+\frac12]=E[X_{n-1}]+\frac12$$