Question 1174870: We have a random variable X = total number of success in n trials.
If X is b(n,p) and assume n is some fixed number and p is to be estimated. Define the random variable p hat = X/n. Show that P hat is an unbiased estimator of p.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Let's break down how to show that p̂ (p-hat) = X/n is an unbiased estimator of p.
**1. Definitions**
* **X:** The total number of successes in n trials.
* **X ~ b(n, p):** X follows a binomial distribution with n trials and probability of success p.
* **p̂ = X/n:** The estimator for p.
* **Unbiased Estimator:** An estimator is unbiased if its expected value is equal to the true parameter it's estimating. In this case, we need to show E[p̂] = p.
**2. Expected Value of X**
* Since X ~ b(n, p), the expected value of X is:
* E[X] = np
**3. Expected Value of p̂**
* We want to find E[p̂] = E[X/n].
* Using the linearity of expectation, we can write:
* E[p̂] = E[X/n] = (1/n) * E[X]
**4. Substitute E[X]**
* Substitute E[X] = np into the equation:
* E[p̂] = (1/n) * (np)
**5. Simplify**
* E[p̂] = (np)/n
* E[p̂] = p
**Conclusion**
Since E[p̂] = p, we have shown that p̂ = X/n is an unbiased estimator of p. This means that, on average, the sample proportion (p̂) will equal the true population proportion (p).
|
|
|
