https://file.aiquickdraw.com/mathbot/file/17365067514755d6fvbs1.jpg

This is a statistical problem about proving that an estimator \( t \) is unbiased for the population variance \( \sigma^2 \) and calculating its variance. Let me guide you step by step.

---

### Problem Restatement:
Given:
- Independent measurements \( x[n] \) that follow a normal distribution \( \mathcal{N}(\mu, \sigma^2) \), where \( n = 0, 1, \ldots, N-1 \).
- An estimator for variance is defined as:
  \[
  t = \frac{1}{N} \sum_{n=0}^{N-1} \left( x[n] - \mu \right)^2
  \]

We are asked to:
1. **Prove that \( t \) is an unbiased estimator of \( \sigma^2 \).**
2. **Calculate the variance of the estimator \( t \).**

---

### 1. **Proving Unbiasedness of \( t \):**

The estimator \( t \) is unbiased if:
\[
\mathbb{E}[t] = \sigma^2
\]

#### Steps:
1. Substitute the definition of \( t \):
   \[
   \mathbb{E}[t] = \mathbb{E}\left[\frac{1}{N} \sum_{n=0}^{N-1} \left( x[n] - \mu \right)^2 \right]
   \]

2. Use the linearity of expectation:
   \[
   \mathbb{E}[t] = \frac{1}{N} \sum_{n=0}^{N-1} \mathbb{E}\left[ \left( x[n] - \mu \right)^2 \right]
   \]

3. Recall the definition of variance:
   For \( x[n] \sim \mathcal{N}(\mu, \sigma^2) \),
   \[
   \mathbb{E}\left[ (x[n] - \mu)^2 \right] = \text{Var}(x[n]) = \sigma^2
   \]

4. Substitute this result:
   \[
   \mathbb{E}[t] = \frac{1}{N} \sum_{n=0}^{N-1} \sigma^2
   \]

5. Simplify the sum:
   \[
   \mathbb{E}[t] = \frac{1}{N} \cdot N \cdot \sigma^2 = \sigma^2
   \]

Thus, \( \mathbb{E}[t] = \sigma^2 \), which proves that \( t \) is an unbiased estimator of \( \sigma^2 \).

---

### 2. **Variance of the Estimator \( t \):**

The variance of \( t \) is defined as:
\[
\text{Var}(t) = \mathbb{E}[t^2] - (\mathbb{E}[t])^2
\]

#### Steps:
1. Expand \( t \):
   \[
   t = \frac{1}{N} \sum_{n=0}^{N-1} (x[n] - \mu)^2
   \]

   Denote \( y_n = (x[n] - \mu)^2 \). Then:
   \[
   t = \frac{1}{N} \sum_{n=0}^{N-1} y_n
   \]

2. Recall the variance formula for the mean:
   If \( y_n \) are independent and identically distributed random variables, then:
   \[
   \text{Var}\left( \frac{1}{N} \sum_{n=0}^{N-1} y_n \right) = \frac{1}{N^2} \sum_{n=0}^{N-1} \text{Var}(y_n)
   \]

3. Find \( \text{Var}(y_n) \):
   Since \( y_n = (x[n] - \mu)^2 \), use the properties of the normal distribution to calculate higher moments. Specifically, for \( x[n] \sim \mathcal{N}(\mu, \sigma^2) \), it can be shown that:
   \[
   \mathbb{E}[y_n^2] = \mathbb{E}[(x[n] - \mu)^4] = 3\sigma^4
   \]
   Thus:
   \[
   \text{Var}(y_n) = \mathbb{E}[y_n^2] - (\mathbb{E}[y_n])^2 = 3\sigma^4 - \sigma^4 = 2\sigma^4
   \]

4. Substitute back into the variance formula:
   \[
   \text{Var}(t) = \frac{1}{N^2} \cdot N \cdot 2\sigma^4 = \frac{2\sigma^4}{N}
   \]

---

### Final Results:

1. **Unbiasedness:**
   \[
   \mathbb{E}[t] = \sigma^2
   \]

2. **Variance of \( t \):**
   \[
   \text{Var}(t) = \frac{2\sigma^4}{N}
   \]

---

Do you want further clarifications or detailed derivations? Let me know! Here are some additional related questions:

---

### Related Questions:
1. What changes if \( \mu \) is unknown and replaced by the sample mean?
2. How is \( t \) related to the sample variance \( S^2 \)?
3. Can \( t \) be considered a maximum likelihood estimator (MLE)?
4. How would the results change for a non-normal distribution?
5. What are the implications of \( N \to \infty \) on the variance of \( t \)?

---

### Tip:
When working with estimators, always check both unbiasedness and variance to understand the trade-off between accuracy and variability.

Δείξτε ότι ο εκτιμητής t είναι αμερόληπτος για τη διασπορά σ^2 και να υπολογιστεί η διασπορά του εκτιμητή t. Ο εκτιμητής ορίζεται ως: t = (1/N) * Σ (x[n] - μ)^2 για n = 0, 1, ..., N-1.

Learn how to prove that the estimator t is unbiased for the population variance σ^2 and calculate its variance. This solution includes key statistical concepts, such as unbiased estimators, properties of variance, and moments of normal distributions. Ideal for university-level statistics courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation