p-vector uniform distribution in ball and $X_i$ i.i.d exponential distribution ($\lambda$) with $ \theta = E\{ X_1 -t ∣ X_1 \gt t \}$

Question

There's some questions:

First one:

Suppose $X_1, \dots, X_n$ are p-vector uniform distribution in the ball $B_\theta = \{x ∣ \Vert x \Vert \lt \theta \} ;\theta>0 $ is an unknown parameter.

(a) Find the MLE of $\theta$. denote $\hat\theta_n$

(b) Find the asymptotic distribution of $ n (\theta - \hat\theta_n )$

Second question:

Suppose that $X_1, \dots, X_n$ i.i.d random variable follows exponential distribution ($\lambda$) and let $ \theta = E\{ X_1 -t ∣ X_1 \gt t \}$

(a) Find the MLE of $\theta$. denote $\hat\theta_n$

(b) Find the asymptotic distribution of $ {\sqrt n} (\hat\theta_n - \theta )$

In problem one, I have no idea about how to solve the question.

In problem two, I knows $ \theta = \{ X_1 -t ∣ X_1 \gt t \}$ should follows Exponential distribution with ($n\lambda$) since the memoryless property and order statistic of $X_1$. But with $ \theta = E\{ X_1 -t ∣ X_1 \gt t \}$ , I don't know how to compute its MLE? Isn't it (that is , $\theta$) just a value since it's a expectation value of $X_1$ , that is, $E \{ X_1 \} $?

Thanks.