Question

Xis a random variable taking values O and I with respective probabilities q and p. A
sample X1,X2, •••• ,X. of size n is taken from the distribution. If r = I,x,, shows that
isl
the bias of the estimator tends to zero as n tends to infinity

Answer 1

Answer to Question #74983, Math / Statistics and Probability

$X$ is a random variable taking values 0 and 1 with respective probabilities $q$ and $p$. A sample $X_{1}, X_{2}, \ldots, X$ of size $n$ is taken from the distribution. If $r = 1, x$, show that the bias of the estimator tends to zero as $n$ tends to infinity.

Solution.

We have Bernoulli distribution with the probability mass function:

$$f(x_i, p) = p^{x_i} (1 - p)^{1 - x_i} \quad \text{for } x_i \in \{0, 1\}$$

Then the bias of the estimator:

$$B(\hat{p}) = E(\hat{p}) - p$$

With estimator:

$$\hat{p} = \frac{\sum_{i=1}^{n} x_i}{n}$$

Variance of estimator:

$$\operatorname{var}(\hat{p}) = \operatorname{var}\left(\frac{\sum_{i=1}^{n} x_i}{n}\right) = \frac{1}{n^2} \sum_{i=1}^{n} \operatorname{var}(x_i) = \frac{1}{n^2} \sum_{i=1}^{n} (pq) = \frac{npq}{n^2} = \frac{pq}{n}$$ $$\lim_{n \to \infty} \operatorname{var}(\hat{p}) = \lim_{n \to \infty} \frac{pq}{n} = 0$$

Answer provided by https://www.AssignmentExpert.com