The problem states that we have a sample of $n$ independent and identically distributed (i.i.d.) random variables $X_1, \dots, X_n$, each following a Bernoulli distribution with parameter $p$, denoted $X_i \sim \text{Ber}(p)$. The task is to compute the biases for different estimators of $p$.

Let me outline the approach to solve this:

1. Expectation and Bias:

   • Suppose we have an estimator $\hat{p}$ for $p$.
   • The bias of $\hat{p}$ is defined as: $\text{Bias}(\hat{p}) = \mathbb{E}[\hat{p}] - p$ where $\mathbb{E}[\hat{p}]$ is the expected value of $\hat{p}$.

2. Common Estimators for $p$: Without seeing the specific estimators listed in the problem, typical estimators for the parameter $p$ in a Bernoulli distribution include:

   • The sample mean $\hat{p}_1 = \frac{1}{n} \sum_{i=1}^n X_i$.
   • Any other biased estimator (such as transformations or scaled versions of the sample mean).

3. Computing Bias:

   • For the sample mean estimator $\hat{p}_1$: $\mathbb{E}\left(\hat{p}_1\right) = \mathbb{E}\left(\frac{1}{n} \sum_{i=1}^n X_i\right) = \frac{1}{n} \sum_{i=1}^n \mathbb{E}(X_i) = \frac{1}{n} \cdot n \cdot p = p$ Thus, the bias of $\hat{p}_1$ is $\text{Bias}(\hat{p}_1) = \mathbb{E}[\hat{p}_1] - p = 0$.

To proceed, please let me know if specific estimators are provided in the question, and I'll calculate the biases for each. Alternatively, I can outline common estimators and their biases if needed.

Would you like further details on this approach, or do you have any additional questions? Here are some related questions for exploration:

1. How does the sample size $n$ affect the bias of different estimators?
2. What other properties of an estimator (like variance) are important besides bias?
3. Why is the sample mean an unbiased estimator for the Bernoulli parameter $p$?
4. How would the bias calculation change if the estimator were nonlinear in $X_i$?
5. Can bias correction techniques make an estimator unbiased?

Tip: Remember, bias is just one measure of an estimator's accuracy; understanding its variance is also crucial for assessing its reliability.