Question 1

Consider the following input data points: | x | $y$ | |---|---| | $[2, 3]$ | 5 | | $[-1, 1]$ | 2 | | $[4, 2]$ | 7 | | $[0, -2]$ | 1 | | $[-3, 5]$ | 4 | Suppose we fit a linear model $f(\mathrm{x}) = x_1 + 2x_2$, where $\mathrm{x} = (x_1, x_2)$. Compute the value of the loss function $L$ for this dataset which is defined as $L = \dfrac{1}{n} \sum\limits_{i=1}^{n} (f(\mathrm{x}^i) - y^i)^2$. Let $X$ and $Y$ be two independent random variables, where $X \sim \mathrm{Normal}(2, 5)$ and $Y \sim \mathrm{Normal}(5, 9)$. Define $Z = 3X - 2Y$. Find the value of $P(Z > 8)$. Enter the answer correct to three decimal places. **Hint:** Use the following values of $F_Z$ if required: - $F_Z(1.33) = 0.90824$ - $F_Z(-1.33) = 0.09176$ - $F_Z(2.088) = 0.98169$ - $F_Z(-2.088) = 0.01831$ Figure from the original question paper