Final answer:
The estimator x+1/n+2 is a biased estimator of the binomial parameter θ because its expected value is θ + 3/n, which is not equal to θ unless n tends to infinity. However, it is asymptotically unbiased as the bias approaches 0 when n increases.
Step-by-step explanation:
The question deals with the concept of estimating a binomial parameter θ, where θ is traditionally represented by p, the probability of success in a binomial distribution. An estimator is called biased if its expected value does not equal the parameter it is estimating. To show whether the estimator x+1/n+2 is biased, we need to calculate its expected value and compare it to the actual parameter θ.
For a binomial distribution X ~ B(n, θ), the expected value E(X) equals nθ. Using this, the expected value of the estimator would be E(x+1/n+2) = E(x)/n + 3/n. Since E(x) = nθ, this simplifies to θ + 3/n, which does not equal to θ unless n tends to infinity.
This means that x+1/n+2 is indeed a biased estimator of θ when n is finite, since its expected value is not equal to θ. However, as n increases, the bias, which is 3/n, tends to 0. Therefore, x+1/n+2 is asymptotically unbiased because the bias vanishes as n approaches infinity, making its expected value converge to the true parameter θ.