Questão 1

Mostre que \(\bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i\) é um estimador não viciado para \(\mu\).

\[ \begin{align*} E(\bar{X}) &= E\left(\frac{1}{n} \sum_{i=1}^{n} X_i\right) \\ \\ E(\bar{X}) &= \frac{1}{n} E\left(\sum_{i=1}^{n} X_i\right) \\ \\ E(\bar{X}) &= \frac{1}{n} \sum_{i=1}^{n} E(X_i) \\ \\ E(\bar{X}) &= \frac{1}{n} \cdot n \cdot \mu \\ \\ \hat{\mu} &= \bar{X} \end{align*} \]

Questão 2

Mostre que \(S^2 \ = \frac{1}{n \ - \ 1} \sum_{i = 1}^{n}(X_I \ - \ \bar{X})^2\) é um estimador não viciado para \(\sigma^2\)

\[ \begin{align*} E(S^2) &= E\left(\frac{1}{n - 1} \sum_{i = 1}^{n}(X_i - \bar{X})^2\right) \\ \\ E(S^2) &= \frac{1}{n - 1} E\left(\sum_{i = 1}^{n}(X_i - \bar{X})^2\right) \\ \\ E(S^2) &= \frac{1}{n - 1} E\left(\sum_{i = 1}^{n}(X_i^2 - 2X_i \bar{X} + \bar{X}^2)\right) \\ \\ E(S^2) &= \frac{1}{n - 1} \left[E\left(\sum_{i = 1}^{n}X_i^2\right) - 2 E\left(\sum_{i = 1}^{n} X_i \bar{X}\right) + E\left(n \bar{X}^2\right)\right] \\ \\ E(S^2) &= \frac{1}{n - 1} \left[\sum_{i = 1}^{n} E(X_i^2) - 2 \sum_{i = 1}^{n} E(X_i) E(\bar{X}) + n E(\bar{X}^2)\right] \\ \\ E(S^2) &= \frac{1}{n - 1} \left[\sum_{i = 1}^{n} E(X_i^2) - 2 n \bar{X}^2 + n \bar{X}^2\right] \\ \\ E(S^2) &= \frac{1}{n - 1} \left[\sum_{i = 1}^{n} E(X_i^2) - n \bar{X}^2\right] \\ \\ E(S^2) &= \frac{1}{n - 1} \left[n E(X_i^2) - n E(\bar{X}^2)\right] \end{align*} \]

Sabe-se que: \[ \begin{align*} \text{Var}(\bar{X}) &= E(\bar{X}^2) - E(\bar{X})^2 \\ \\ E(\bar{X}^2) &= \text{Var}(\bar{X}) + E(\bar{X})^2 \\ \\ E(\bar{X}^2) &= \frac{\sigma^2}{n} + \mu^2 \end{align*} \]

agora temos: \[ \begin{align*} \text{Var}(X_i) &= E(X_i^2) - E(X_i)^2 \\ \\ E(X_i^2) &= \text{Var}(X_i) + E(X_i)^2 \\ \\ E(X_i^2) &= \sigma^2 + \mu^2 \end{align*} \]

Substituindo:

\[ \begin{align*} E(S^2) &= \frac{1}{n - 1} \cdot n \cdot (\sigma^2 + \mu^2) - n \cdot \left(\frac{\sigma^2}{n} + \mu^2\right) \\ \\ E(S^2) &= \frac{1}{n - 1} \cdot n \sigma^2 + n \mu^2 - \sigma^2 - n \mu^2 \\ \\ E(S^2) &= \frac{1}{n - 1} \cdot n \sigma^2 - \sigma^2 \\ \\ E(S^2) &= \frac{n \sigma^2 - \sigma^2}{n - 1} \\ \\ E(S^2) &= \frac{(n - 1) \sigma^2}{n - 1} \\ \\ E(S^2) &= \hat{\sigma}^2 \end{align*} \]