The given problem involves a standard normal distribution on $\mathbb{R}^2$, where $Z = (Z_1, Z_2)$ consists of two independent standard normal variables $Z_1, Z_2 \sim \mathcal{N}(0,1)$. Let's solve each part step by step.

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Part (a): Show that $\|Z\|^2$ is exponentially distributed with parameter $\frac{1}{2}$.

Step 1: Definition of $\|Z\|^2$:

The norm squared of $Z$ is: $\|Z\|^2 = Z_1^2 + Z_2^2.$ Here, $Z_1$ and $Z_2$ are independent standard normal random variables.

Step 2: Distribution of $Z_1^2$ and $Z_2^2$:

For a standard normal variable $Z_i \sim \mathcal{N}(0,1)$, the square $Z_i^2$ follows a chi-squared distribution with 1 degree of freedom: $Z_i^2 \sim \chi^2(1).$ Since $Z_1$ and $Z_2$ are independent, their sum $Z_1^2 + Z_2^2$ follows a chi-squared distribution with $2$ degrees of freedom: $\|Z\|^2 = Z_1^2 + Z_2^2 \sim \chi^2(2).$

Step 3: Connection between $\chi^2(2)$ and the exponential distribution:

It is known that a chi-squared distribution with $2$ degrees of freedom is equivalent to an exponential distribution with rate parameter $\lambda = \frac{1}{2}$. Specifically: $\|Z\|^2 \sim \text{Exponential}\left(\frac{1}{2}\right).$

Thus, we have shown that $\|Z\|^2$ is exponentially distributed with parameter $\frac{1}{2}$.

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Part (b): Determine the $2 \times 2$ matrices $A$ such that $AZ$ is standard normal in $\mathbb{R}^2$.

Step 1: Standard normal vector properties:

A vector $W \in \mathbb{R}^2$ is standard normal if its components are independent and identically distributed as $\mathcal{N}(0,1)$.

The transformation of $Z$ by a $2 \times 2$ matrix $A$ is given as: $W = AZ,$ where $Z$ is standard normal.

Step 2: Covariance structure of $AZ$:

The covariance matrix of $Z$ is the identity matrix $I_2$, because $Z_1$ and $Z_2$ are independent and have variance 1. For the transformed vector $W = AZ$, the covariance matrix is: $\Sigma_W = A A^\top,$ where $A^\top$ is the transpose of $A$.

For $W$ to be standard normal, its covariance matrix must also equal the identity matrix $I_2$. Thus, we require: $A A^\top = I_2.$

Step 3: Condition for $A$:

The condition $A A^\top = I_2$ means that $A$ is an orthogonal matrix. A matrix $A$ is orthogonal if its rows (or columns) are orthonormal vectors. Formally: $A^\top A = A A^\top = I_2.$

Conclusion:

The $2 \times 2$ matrices $A$ that satisfy $AZ$ being standard normal on $\mathbb{R}^2$ are the orthogonal matrices. These matrices preserve the norm and structure of the vector $Z$.

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Summary of Results:

(a) $\|Z\|^2$ is exponentially distributed with parameter $\frac{1}{2}$.
(b) The matrices $A$ are the $2 \times 2$ orthogonal matrices.

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Let me know if you need a more detailed explanation of any part or further examples!

Here are 5 related questions:

1. What is the chi-squared distribution, and how does it arise from normal random variables?
2. How can the exponential distribution be derived from the chi-squared distribution with 2 degrees of freedom?
3. What are the properties of orthogonal matrices in linear algebra?
4. How does a linear transformation $AZ$ affect the covariance matrix of a multivariate normal vector?
5. Can you generalize this result for $n$-dimensional standard normal vectors?

Tip: Orthogonal matrices preserve lengths and angles, making them fundamental in transformations involving multivariate normal distributions.