MLF End Term: 24 December 2023, Set FDD1 (September 2023 term)

Consider the following input data points:

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:

• $F_Z(1.33) = 0.90824$
• $F_Z(-1.33) = 0.09176$
• $F_Z(2.088) = 0.98169$
• $F_Z(-2.088) = 0.01831$