Explain variance reduction in random forests

Consider a random forest (or bagged ensemble) that predicts at a fixed input \(x\) by averaging \(B\) tree predictions: \[ \hat f_B(x) = \frac{1}{B}\sum_{b=1}^B T_b(x). \] Assume each tree prediction has the same variance \(\sigma^2\) and any pair of tree predictions has correlation \(\rho\). 1. Derive the variance of the ensemble prediction \(\hat f_B(x)\). 2. Explain how this connects to the variance formula for the average of correlated random variables. 3. Interpret the result for the cases \(\rho=0\), \(\rho=1\), and \(B \to \infty\). 4. Based on this result, explain why random forests use bagging and random feature subsampling, and discuss the bias-variance trade-off when making trees more random.