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In statistical estimation, an unbiased estimator is a tool or formula used to estimate a population parameter. The key feature of an unbiased estimator is that its expected value equals the true parameter of the population. This means that on average, the estimator neither overestimates nor underestimates the parameter.In the context of the given problem, the estimator \( \hat{p} = \frac{r-1}{X+r-1} \) is used to estimate the probability \( p \) of success in Bernoulli trials, where \( r \) is the number of required successes and \( X \) is the number of failures.

• An unbiased estimator ensures that \( E(\hat{p}) = p \), guaranteeing accuracy over many trials.
• The importance of unbiased estimators lies in their reliability for making statistical inferences.

Building unbiased estimators often involves manipulating expectations mathematically, as shown in the exercise solution, where \( E(\hat{p}) \) simplifies to \( p \). This proves the estimator's unbiased nature.

The probability mass function (pmf) defines the probabilities of a discrete random variable taking on specific values. For a negative binomial random variable, the pmf determines the probability of observing a certain number of failures before achieving a fixed number of successes in a series of Bernoulli trials.The pmf of a negative binomial distribution is given as:\[P(X = x) = \binom{x + r - 1}{x} p^r (1-p)^x\]where:

• \( x \) is the number of failures.
• \( r \) is the target number of successes.
• \( p \) is the probability of success on each trial.

Understanding the pmf helps us to calculate the likelihood of different outcomes and is a crucial step in defining the expected value and verifying the unbiasedness of estimators. It allows for a clear statistical framework in predicting outcomes in stochastic processes such as repeated trials.

Bernoulli trials refer to the simplest type of probabilistic experiments, where there are only two possible outcomes: success or failure. Each trial is independent, meaning the outcome of one does not influence the others.In the context of the negative binomial distribution:

• Each trial's probability of success is \( p \), while failure is \( 1-p \).
• The trials are repeated until a specified number of successes \( r \) is achieved.

Bernoulli trials form the basis for more complex distributions, such as the binomial and negative binomial distributions. The sequence of successes and failures observed in a real-life scenario, like interviewing people, follows this principle.For example, the given problem's sequence "SFFSFFFSSS" illustrates this process of Bernoulli trials, where successive failures occur before reaching the desired number of successes. This exemplifies the practical application of these theoretical trials.

The expected value is a fundamental concept in probability, representing the long-term average or mean of a random variable's possible outcomes, based on its probability distribution. Essentially, it's the weighted average where every possible outcome is multiplied by its probability of occurrence.For the estimator \( \hat{p} = \frac{r-1}{X+r-1} \), the expected value helps us establish its unbiasedness:

• We calculate \( E(\hat{p}) \) to compare its result with \( p \).
• The simplified calculation shows that \( E(\hat{p}) = p \), confirming \( \hat{p} \) is unbiased.

Understanding expected value enables better prediction and decision-making under uncertainty. In statistical estimations like our problem, it is vital for verifying whether an estimator accurately targets a true parameter. In practical scenarios, it bridges theoretical findings with real-world data through its predictive power.