Question #3 Let $X_1, X_2, \dots, X_n$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. Prove that $\bar{X}$ and $S^2$ are independent random variables. To get credit, show each step of the proof clearly.
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Step 1: We need to show that the covariance between X and S² is zero. Show more…
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Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample from a normal distribution $N\left(\mu, \sigma^{2}\right)$. Show that $$ \sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2}=\sum_{i=2}^{n}\left(X_{i}-\bar{X}^{\prime}\right)^{2}+\frac{n-1}{n}\left(X_{1}-\bar{X}^{\prime}\right)^{2}, $$ where $\bar{X}=\sum_{i=1}^{n} X_{i} / n$ and $\bar{X}=\sum_{i=2}^{n} X_{i} /(n-1)$ Hint: Replace $X_{i}-\bar{X}$ by $\left(X_{i}-\bar{X}^{\prime}\right)-\left(X_{1}-\bar{X}\right) / n .$ Show that $\sum_{i=2}^{n}\left(X_{i}-\bar{X}^{\prime}\right)^{2} / \sigma^{2}$ has a chi-square distribution with $n-2$ degrees of freedom. Prove that the two terms in the right-hand member are independent. What then is the distribution of $$ \frac{[(n-1) / n]\left(X_{1}-\bar{X}^{\prime}\right)^{2}}{\sigma^{2}} ? $$
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Suppose X1, X2, ..., Xn are i.i.d. normal random variables, each with mean μ and variance σ². Let ¯X denote the sample mean and let S² = ∑ᵁᵢ₌₁(Xᵢ − ¯X)²/(n − 1). We proved in class that ¯X is normal with mean μ and variance σ²/n, (n − 1)S²/σ² is chi-square with (n − 1) degrees of freedom, and that ¯X and S² are independent. Assuming the above facts, show that the random variable (¯X − μ) / (S/∑n) has a t-distribution with (n − 1) degrees of freedom.
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Suppose that $Y_{1}, Y_{2}, \ldots, Y_{n}$ is a random sample from a normal distribution with mean $\mu$ and variance $\sigma^{2}$. The independence of $\sum_{i=1}^{n}\left(Y_{i}-\bar{Y}\right)^{2}$ and $Y$ can be shown as follows. Define an $n \times n$ matrix A by and notice that $\mathbf{A}^{\prime} \mathbf{A}=\mathbf{I}$, the identity matrix. Then, $$\sum_{i=1}^{n} Y_{i}^{2}=\mathbf{Y}^{\prime} \mathbf{Y}=\mathbf{Y}^{\prime} \mathbf{A}^{\prime} \mathbf{A} \mathbf{Y}$$ where $Y$ is the vector of $Y_{i}$ values. a. Show that $$\mathbf{A Y}=\left[\begin{array}{c} \bar{Y} \sqrt{n} \\ U_{1} \\ U_{2} \\ \vdots \\ U_{n-1} \end{array}\right]$$ where $U_{1}, U_{2}, \ldots, U_{\mathrm{n}-1}$ are linear functions of $Y_{1}, Y_{2}, \ldots, Y_{n} .$ Thus, $$\sum_{i=1}^{n} Y_{i}^{2}=n \bar{Y}^{2}+\sum_{i=1}^{n-1} U_{i}^{2}$$ b. Show that the linear functions $Y \sqrt{n}, U_{1}, U_{2}, \ldots, U_{n-1}$ are pairwise orthogonal and hence independent under the normality assumption. (See Exercise $5.130 .$ ) c. Show that $$\sum_{i=1}^{n}\left(Y_{i}-\bar{Y}\right)^{2}=\sum_{i=1}^{n-1} U_{i}^{2}$$ and conclude that this quantity is independent of $Y$. d. Using the results of part (c), show that $$\frac{\sum_{i=1}^{n}\left(Y_{i}-Y\right)^{2}}{\sigma^{2}}=\frac{(n-1) S^{2}}{\sigma^{2}}$$ has a $\chi^{2}$ distribution with $(n-1)$ df.
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