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The problem provided describes a continuous-time random process \( X(t) = A \cos(\omega t) \), where \( t \in \mathbb{R} \), \( \omega \) is a constant, and \( A \) is a standard normal random variable. The task is to find the joint distribution function and the probability density function of \( X(t) \) at two different times.

### Solution Outline
1. **Random Variable Definition**:
   - \( A \) is a standard normal random variable, so \( A \sim \mathcal{N}(0, 1) \).
   - This means \( X(t) = A \cos(\omega t) \) is a Gaussian process, as \( X(t) \) is a linear transformation of \( A \).

2. **Two-Dimensional Distribution**:
   - We need to evaluate \( X(t) \) at two different times, say \( t_1 \) and \( t_2 \).
   - Define \( X(t_1) = A \cos(\omega t_1) \) and \( X(t_2) = A \cos(\omega t_2) \).
   - Let \( X_1 = X(t_1) \) and \( X_2 = X(t_2) \). Then:
     \[
     X_1 = A \cos(\omega t_1), \quad X_2 = A \cos(\omega t_2)
     \]
   - \( X_1 \) and \( X_2 \) are jointly Gaussian because they are linear functions of the Gaussian variable \( A \).

3. **Mean and Covariance**:
   - The mean of \( X_1 \) and \( X_2 \) is zero, as \( A \) has a mean of zero:
     \[
     \mathbb{E}[X_1] = \mathbb{E}[A] \cdot \cos(\omega t_1) = 0, \quad \mathbb{E}[X_2] = \mathbb{E}[A] \cdot \cos(\omega t_2) = 0
     \]
   - The variances are:
     \[
     \text{Var}(X_1) = \mathbb{E}[X_1^2] = \cos^2(\omega t_1), \quad \text{Var}(X_2) = \mathbb{E}[X_2^2] = \cos^2(\omega t_2)
     \]
   - The covariance between \( X_1 \) and \( X_2 \) is:
     \[
     \text{Cov}(X_1, X_2) = \mathbb{E}[X_1 X_2] = \mathbb{E}[A^2] \cos(\omega t_1) \cos(\omega t_2) = \cos(\omega t_1) \cos(\omega t_2)
     \]

4. **Joint Probability Density Function**:
   - Since \( (X_1, X_2) \) are jointly Gaussian with zero mean, their joint probability density function (PDF) can be written as:
     \[
     f_{X_1, X_2}(x_1, x_2) = \frac{1}{2 \pi \sigma_{X_1} \sigma_{X_2} \sqrt{1 - \rho^2}} \exp \left( -\frac{1}{2(1 - \rho^2)} \left( \frac{x_1^2}{\sigma_{X_1}^2} + \frac{x_2^2}{\sigma_{X_2}^2} - 2 \rho \frac{x_1 x_2}{\sigma_{X_1} \sigma_{X_2}} \right) \right)
     \]
   where:
   - \( \sigma_{X_1} = \cos(\omega t_1) \),
   - \( \sigma_{X_2} = \cos(\omega t_2) \),
   - \( \rho = \frac{\text{Cov}(X_1, X_2)}{\sigma_{X_1} \sigma_{X_2}} = \frac{\cos(\omega t_1) \cos(\omega t_2)}{\cos(\omega t_1) \cos(\omega t_2)} = 1 \).

Thus, \( X(t) \) at two distinct times forms a degenerate Gaussian distribution if \( t_1 \) and \( t_2 \) are such that \( \cos(\omega t_1) = \cos(\omega t_2) \), or a bivariate Gaussian distribution otherwise.

Would you like a more detailed breakdown, or have specific questions about this process?

### Related Questions:
1. How does the value of \( \omega \) affect the covariance between \( X_1 \) and \( X_2 \)?
2. What would the joint distribution look like if \( A \) were not a standard normal variable?
3. Can we generalize this approach for more than two time points?
4. How does the process change if \( A \) has a mean other than zero?
5. What properties does \( X(t) \) have as a Gaussian process?

**Tip**: Understanding Gaussian processes and how linear transformations affect Gaussian variables is fundamental in stochastic processes.

有连续时间随机过程 X(t) = A cos(ωt), t ∈ R, ω 是一个正常数, A 是一个标准正态一维随机变量. 试求 X(t) 的一维、二维概率分布函数与概率密度函数。

Explore how to determine the one-dimensional and two-dimensional probability distribution functions and probability density functions of the process X(t) = A cos(ωt), where A is a standard normal random variable. This problem involves Gaussian processes, joint probability distributions, and probability density functions.

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