Let $\quad f: \mathbb{R} \rightarrow \mathbb{R} \quad$ be such that $|f(x)-f(y)| \leq(x-y)^2$ for all $x, y \in \mathbb{R}$.

Then $\quad f(1)-f(0)=$ ____________

A bag contains 5 white balls and 10 black balls. In a random experiment, $n$ balls are drawn from the bag one at a time with replacement. Let $S_n$ denote the total number of black balls drawn in the experiment.

The expectation of $S_{100}$ denoted by $E\left[S_{100}\right]=$ ___________ (Round off to one decimal place)

Consider a directed graph $G=(V, E)$, where $V=\{0,1,2, \ldots, 100\}$ and $E=\{(i$, $j): 0 < j-i \leq 2$, for all $i, j \in V\}$. Suppose the adjacency list of each vertex is in decreasing order of vertex number, and depth-first search (DFS) is performed at vertex 0 . The number of vertices that will be discovered after vertex 50 is___________

Consider designing a linear classifier

$$ y=\operatorname{sign}(f(x ; w ; b)), f(x ; w, b)=w^{\mathrm{T}} x+b $$

on a dataset

$$ \begin{aligned} & D=\left\{\left(x_1, y_1\right),\left(x_2, y_2\right) \ldots \ldots\left(x_N, y_N\right)\right\} \\ & x_i \in \mathbb{R}^d, y_i \in\{+1,-1\}, i=1,2, \ldots \ldots, N \end{aligned} $$

Recall that the sign function outputs +1 if the argument is positive, and -1 if the argument is non-positive. The parameters $w$ and $b$ are updated as per the following training algorithm:

$$ w_{\text {new }}=w_{\text {old }}+y_n x_n, b_{\text {new }}=b_{\text {old }}+y_n $$

Whenever sign $\left(f\left(x_n ; w_{\text {old }}, b_{\text {old }}\right)\right) \neq y_n$ In other words, whenever the classifier wrongly predicts a sample $\left(x_n, y_n\right)$ from the dataset, $w_{\text {old }}$ gets updated to $w_{\text {new }}$, and likewise $b_{\text {old }}$ gets updated to $b_{\text {new }}$. Consider the case

$$ \left(x_n,+1\right), f\left(x_n ; w_{\text {old }}, b_{\text {old }}\right)<0 \text {. Then } $$