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.