Let $X_1,\dots,X_n$ be i.i.d. from the following probability distribution
$$ f(x) = \begin{cases} 0, & \text{with probability $p$} \\ \operatorname{Uniform}[0,\theta], & \text{with probability $1-p$} \end{cases}$$
Assume that $p$ is a known constant in $(0,1)$ and that $\theta>0$ is the parameter of interest.
1) Based only on one observation, $X_1$, find all the unbiased estimators for $\theta$.
2) Write the joint likelihood for $X_1,\dots,X_n$.
So far, I have found the only unbiased estimator to be $\frac{2X_1}{1-p}$. I am struggling with the joint distribution. I originally thought of this as a conditional distribution given a Bernoulli random variable, but I cannot get the joint distribution to work out.
Since you are told the $X_i$ are iid, the joint cdf is $$ F_{X_1,\ldots,X_n}(x_1, \ldots, x_n) = \mathbb P(X_1 \le x_1, \ldots, X_n \le x_n) = \prod_{i=1}^n F_{X_1}(x_i)$$