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In probability theory and statistics, an estimator is a rule or formula that allows us to make deductions about unknown parameters based on observed data. An unbiased estimator is a special kind of estimator with a unique property: its expected value is equal to the parameter it estimates. Using the example of \(P = X / n\), we're trying to estimate the probability \(p\) of success in a binomial distribution. The estimation procedure is unbiased because the expected value of \(P\), denoted as \(E(P)\), equals \(p\). For the binomial distribution, the expectation of the random variable \(X\) is given by:

• \(E(X) = np\)

To find \(E(P)\), we evaluate it as follows:

• \(E(P) = E(X/n) = (1/n) \cdot E(X) = (1/n) \cdot np = p\)

This shows that the average or expected value of \(P\) is indeed \(p\). Hence, \(P\) is an unbiased estimator of \(p\). By relying on unbiased estimators, we ensure that on average, our estimates are accurate.

When an estimator's expected value does not equal the parameter it tries to estimate, it is called a biased estimator. The presence of bias means that there is an inherent systematic error in the estimation process. Take, for instance, \(P' = \frac{X + \sqrt{n}/2}{n + \sqrt{n}}\), which is a supposed estimate of \(p\). To determine if it is biased, we compute its expected value:

• \(E(P') = E \left( \frac{np + \sqrt{n}/2}{n + \sqrt{n}} \right)\)
• \(= p \cdot \frac{n + \sqrt{n}/2}{n + \sqrt{n}}\)

Simplifying, this does not equal \(p\). The calculation shows that \(E(P')\) results in a value different from \(p\) due to the additional \(\sqrt{n}\) term in both the numerator and the denominator. Thus, \(P'\) is a biased estimator.Bias in an estimator can be intentional for simpler calculations or to reduce variance, but it often requires adjustments to correct the error.

The binomial distribution is a discrete probability distribution. It describes the number of successes in a fixed number of independent experiments, each with the same probability of success. This makes it particularly useful for scenarios where you have two possible outcomes, such as a coin flip. Parameters of the binomial distribution include:

• \(n\): The total number of trials
• \(p\): The probability of success on an individual trial

If \(X\) is a binomial random variable, it represents the number of successes in \(n\) trials with probability \(p\). The outcomes are often modelled in situations like survey responses or natural phenomena. A critical property of the binomial distribution is its expectation and variance, which are given by:

• Expected value: \(E(X) = np\)
• Variance: \( ext{Var}(X) = np(1-p)\)

The binomial distribution is foundational in understanding and applying probabilistic models and is crucial in constructing unbiased and biased estimators alike.