Welch's Two-Sample t-Test and Satterthwaite Approximation

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Hypothesis Testing.

Your Task
Given two independent normal samples $X_1, \dots, X_{n_1} \sim \mathcal{N}(\mu_1, \sigma_1^2)$ and $Y_1, \dots, Y_{n_2} \sim \mathcal{N}(\mu_2, \sigma_2^2)$ where $\sigma_1^2 \ne \sigma_2^2$ are unknown.
(1) State Welch's $t$-statistic for testing $H_0: \mu_1 = \mu_2$.
(2) State the Welch-Satterthwaite equation for the approximate degrees of freedom $\nu$.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.