
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system

Let $(ξ, η)$ be a two-dimensional Gaussian vector with $\mathrm{E} \xi = \mathrm{E} \eta = 0, \mathrm{D} \xi = \mathrm{D} \eta = 1$, and correlation coefficient $\rho$. Establish the equality

$$
\mathrm{P}(\xi > a, \eta > b) - \mathrm{P}(\xi > a) \mathrm{P}(\eta > b) = \int_{0}^{\rho} \phi(a, b, r) \, \mathrm{d}r
$$

where $a, b \in \mathbb{R}, \phi(a, b, r)$ is the density of the bivariate distribution

$$
\mathscr{N}\left(\binom{0}{0}, \begin{pmatrix}
1 & r \\
r & 1
\end{pmatrix} \right)
$$

Using Hoeffding's formula

$$
\operatorname{cov}(f(\xi), f(\eta)) = \int_{\mathbb{R}^{2}} [\mathrm{P}(\xi > a, \eta > b) - \mathrm{P}(\xi > a) \mathrm{P}(\eta > b)] f^{\prime}(a) g^{\prime}(b) \, \mathrm{d}a \, \mathrm{d}b
$$

and the given equality, establish that

$$
\operatorname{cov}(f(\xi), g(\eta)) = \int_{0}^{\rho} \mathrm{E} f^{\prime}\left(\xi_r\right) g^{\prime}\left(\eta_r\right) \, \mathrm{d}r
$$

for such smooth functions $f$ and $g$ that $\mathrm{E}|f(\xi) g(\eta)| < \infty$, where $(\xi_r, \eta_r)$ is a Gaussian vector with density $\phi(a, b, r)$.

Present the answer in LaTex format: \boxed{Your answer}
user