Homework 1#
Section 3.5 (14-17, 22)#
Problem 14#
\[ P(X_1 > X_2 + X_3 + 2) = P(X_1 - X_2 - X_3) \]
\[ Y = X_1 - X_2 - X_3 \]
\[\begin{split}
Y = aX =
\begin{bmatrix}
1 & -1 & -1
\end{bmatrix}
\begin{bmatrix}
X_1 \\ X_2 \\ X_3
\end{bmatrix}
\end{split}\]
Therefore, we see that \(Y \sim \text{Normal}(a \mu, a \Sigma a^T)\).
\[\begin{split}
a \mu =
\begin{bmatrix}
1 & -1 & -1
\end{bmatrix}
\begin{bmatrix}
0 \\ 0 \\ 0
\end{bmatrix}
= 0
\end{split}\]
\[ \begin{align}\begin{aligned}\begin{split}
\begin{align*}
a \Sigma a^T &=
\begin{bmatrix}
1 & -1 & -1
\end{bmatrix}
\begin{bmatrix}
1 & 0 & 0 \\
0 & 2 & 1 \\
0 & 1 & 2
\end{bmatrix}
\begin{bmatrix}
1 \\ -1 \\ -1
\end{bmatrix} \\\end{split}\\\begin{split} &=
\begin{bmatrix}
1 & -3 & -3
\end{bmatrix}
\begin{bmatrix}
1 \\ -1 \\ -1
\end{bmatrix} \\\end{split}\\\begin{split} &= 1 + 3 + 3 \\
&= 7
\end{align*}
\end{split}\end{aligned}\end{align} \]
Therefore, \(Y \sim \text{Normal}(0, 7)\), and so
\[\begin{split}
\begin{align*}
P(Y > 2) &= 1 - P(Y \leq 2) \\
&= 1 - P(Z \leq \frac{2-0}{\sqrt{7}}), & Z \sim \text{Normal}(0,1) \\
&\approx 1 - 0.775 \\
&\approx 0.225
\end{align*}
\end{split}\]
Problem 16#
\[\begin{split} Y = aX =
\begin{bmatrix}
1 & 1 \\
1 & -1
\end{bmatrix}
\begin{bmatrix}
X_1 \\ X_2
\end{bmatrix}
=
\begin{bmatrix}
X_1 + X_2 \\
X_1 - X_2
\end{bmatrix}
\end{split}\]
Therefore, \(Y \sim \text{Normal}(a\mu, a\Sigma a^T)\).
To show that \(X_1 + X_2\) and \(X_1 - X_2\) are independent when \(Var(X_1) = Var(X_2)\), we will show that their covariance is \(0\).
\[\begin{split}
\begin{align*}
Var(Y) = a \Sigma a^T &=
\begin{bmatrix}
1 & 1 \\
1 & -1
\end{bmatrix}
\begin{bmatrix}
\sigma_1^2 & \sigma_{1,2} \\
\sigma_{1,2} & \sigma_2^2
\end{bmatrix}
\begin{bmatrix}
1 & 1 \\
1 & -1
\end{bmatrix} \\
&=
\begin{bmatrix}
\sigma_1^2 + \sigma_{1,2} & \sigma_{1,2} + \sigma_2^2 \\
\sigma_1^2 - \sigma_{1,2} & \sigma_{1,2} - \sigma_2^2
\end{bmatrix}
\begin{bmatrix}
1 & 1 \\
1 & -1
\end{bmatrix} \\
&=
\begin{bmatrix}
\sigma_1^2 + 2\sigma_{1,2} + \sigma_2^2 & \sigma_1^2 - \sigma_2^2 \\
\sigma_1^2 - \sigma_2^2 & \sigma_1^2 - 2\sigma_{1,2} + \sigma_2^2
\end{bmatrix}
\end{align*}
\end{split}\]
It follows from here that if \(Var(X_1) = Var(X_2)\), then \(Cov(X_1 + X_2, X_1 - X_2) = 0\) and, therefore, \(X_1 + X_2\) and \(X_1 - X_2\) are independent.