data c140 ## 1. Prediction with Sums Let $X_1, X_2, \ldots, X_n$ be i.i.d. with expectation $\mu$ and variance $\sigma^2$. Let $S = \sum_{i=1}^n X_i$. **a)** Find the least squares predictor of $S$ based on $X_1$, and find the mean squared error (MSE) of the predictor. **b)** Find the least squares predictor of $X_1$ based on $S$, and find the MSE of the predictor. Is the predictor a linear function of $S$? If so, it must also be the best among all linear predictors based on $S$, which is commonly known as the regression predictor. [Consider whether your predictor in (b) would be different if $X_1$ were replaced by $X_2$, or by $X_3$, or by $X_i$ for any fixed $i$. Then use symmetry and the additivity of conditional expectation.]